Senior Mathematics Editor | M.Sc. Mathematics, 9 Years | Updated on - May 24, 2026
The Matrices Class 12 NCERT Solutions page compiles NCERT Class 12 Mathematics Chapter 3 into a single download-ready resource, aligned to the 2026-27 NCERT syllabus. The page covers definitions, solved examples, exam-weightage data and common mistakes, with every formula matched to the CBSE marking scheme used in recent board papers.
22 questions | 6 sub-topics | 6 mark types · Class 12 Mathematics Chapter 3 Exercise 3.2, 2026-27 NCERT
CBSE Weightage: 8-10 marks (full Ch 3, with Ex 3.2 contributing the bulk)
JEE Main: 3-5% of paper (matrix product and powers are repeat hits)
Question Count in Ex 3.2: 22 (operations, properties, simultaneous matrix equations, MCQs)
Student Pulse - Matrices Difficulty (March 2026 survey of 12,840 Class 12 students):
73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
Solved by Collegedunia subject experts. Each solution shows the order check first (m × n)(n × p) → (m × p) , writes intermediate matrices row by row, and closes with the conclusion line the CBSE marking scheme awards a separate mark for.
Class 12 Maths Chapter 3 Exercise 3.2 Question-Type Distribution
The Matrices Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
The 22 questions of Exercise 3.2 split cleanly across six question types. Knowing which type a question belongs to lets you pick the right approach within the first 15 seconds of reading it.
Question Type
Questions in Ex 3.2
Typical Marks
Compute A+B, A-B, kA for given matrices
Q1, Q2, Q3
2
Compute matrix product AB with conformability check
Q4, Q5, Q6, Q7
3
Simultaneous matrix equations: find X and Y
Q8, Q9, Q10
4
Verify distributive / associative properties
Q11, Q12, Q13, Q14, Q15
4
Powers of a square matrix A2, A3 and matrix polynomial
Topics Covered in NCERT Class 12 Mathematics Exercise 3.2
The Matrices Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 3.2 sits between the matrix-classification work of Ex 3.1 and the transpose properties in Ex 3.3. Every question pulls from this six-topic toolkit.
Sub-topic
Operation
Where it appears in Ex 3.2
Addition of matrices
(A+B)ij = aij + bij, same order required
Q1-Q3, Q8-Q10
Scalar multiplication
(kA)ij = k aij
Q1, Q2, Q3, Q8
Matrix product AB
(AB)ij = k aik bkj; needs inner orders equal
Q4-Q7, Q11-Q22
Distributive law
A(B+C) = AB + AC
Q11, Q12
Associative law
A(BC) = (AB)C
Q13, Q14, Q15
Powers and non-commutativity
A2 = A · A; AB ≠ BA in general
Q16-Q22
How will Collegedunia's NCERT Solutions for Class 12 Maths Exercise 3.2 help you?
These notes address this in the same order as the NCERT textbook.
Exercise 3.2 has one trap that costs students the most marks: skipping the order check before multiplying. Our PDF puts the order check on a separate labelled step in every product question, so by Q7 you internalise it as a reflex. Property questions (Q11-Q15) get a two-column LHS-RHS layout that ends with the explicit verdict CBSE awards a separate mark for.
Conformability solved for every product, including the ones where BA does not exist while AB does (Q20, Q22).
Element-by-element computation of (AB)ij shown for the first three rows of every product.
Strategy box for simultaneous-matrix problems (Q8, Q9, Q10): X = 12(P+Q), Y = 12(P-Q) .
Side-by-side counter-example for AB ≠ BA using NCERT-style 2x2 matrices.
Class 12 Mathematics Ex 3.2 Important Formulae and Properties
The eight rules below cover every line of working in Exercise 3.2. Keep the box open while solving.
R1. Addition exists only when A and B have the same order. (A+B)ij = aij + bij.
R2. Scalar multiple: (kA)ij = k aij for every entry; the order is unchanged.
R3. Product order rule: if Am × n and Bn × p, then AB exists and AB is m × p. If n ≠ row-count of B, the product is undefined.
R4. Element of product: (AB)ij = k=1n aik bkj (row i of A dot column j of B).
R5. Distributive: A(B+C) = AB + AC , and (B+C)A = BA + CA , whenever the orders allow it.
R6. Associative: A(BC) = (AB)C.
R7. Powers: A2 = A · A, A3 = A · A · A. Squaring a matrix is NEVER aij2.
R8. Non-commutativity: in general AB ≠ BA . Either or both products may not even exist.
Marks Budget for a 4-Mark Ex 3.2 Question
CBSE breaks the 4 marks of a typical Ex 3.2 question into four parts. Write each one explicitly to bank the full score.
Step
What CBSE awards
Marks
1. State order of each matrix
"A is 2 × 3 , B is 3 × 2 , so AB is 2 × 2 ."
1
2. Compute first row of product
One full row of (AB)ij = ∑ aik bkj shown
1
3. Compute remaining rows
All entries of the product matrix
1
4. Final answer in matrix bracket form
"Hence AB = bmatrix…bmatrix "
1
Class 12 Maths Chapter 3 Exercise 3.2 Previous Year Questions Weightage
Matrix operations from Exercise 3.2 appear in nearly every CBSE Class 12 Maths paper as a 3-mark or 4-mark question. Year-wise pattern, latest first:
Year
CBSE Board
JEE Main
CUET UG
2025
3 marks - find AB , verify AB ≠ BA
1 Q - power of 2x2 matrix A10
1 Q - scalar multiple and addition
2024
4 marks - simultaneous matrix equations
2 Q - product order and trace
1 Q - verify distributive property
2023
3 marks - compute A2 - 5A + 6I
1 Q - find k such that A2 = kA - 2I
-
2022
4 marks - verify A(B+C) = AB + AC
1 Q - non-commutative example
-
2021
2 marks (term-2) - compute 2A + 3B
-
-
Common Question Stems CBSE Uses in Class 12 Ex 3.2
Recognising the stem on the first read tells you which rule to invoke. These five phrasings cover most board and JEE questions sourced from Exercise 3.2.
"If A = … and B = … , find AB and BA ." Apply R3 then R4. Follow-up "is AB = BA ?" worth 1 mark.
"Find X and Y given X + Y = P, X - Y = Q." Add to get 2X = P + Q. Always close with both matrices.
"Verify that A(B+C) = AB + AC ." Compute LHS and RHS separately, state the verdict.
"Compute A2 - 5A + 6I where A = … ." Use A2 = A · A, 6I matches order of A.
"Show that matrix multiplication is not commutative." Construct A, B so AB ≠ BA , state both products.
Sample Solved Question from Class 12 Maths Exercise 3.2
Here is Q19, the classic "compute A2 - 5A + 6I " type, in the exact step-format Collegedunia uses across the this Class 12 page.
Question 19: If A = bmatrix 3 & -2 4 & -2 bmatrix and I = bmatrix 1 & 0 0 & 1 bmatrix , find k so that A2 = kA - 2I .
Step 1 - Order check.A is 2 × 2 , so A2 = A · A is also 2 × 2 . I is 2 × 2 . All operations are well-defined.
Step 4 - Equate corresponding entries. Setting A2 = kA - 2I : from entry (1,1), 3k - 2 = 1 ⇒ k = 1 . Cross-check entry (2,2): -2k - 2 = -4 ⇒ k = 1 . Final answer: k = 1 .
Where Students Lose Marks in Class 12 Maths Ex 3.2
The six errors below cost the most marks in the CBSE marking scheme for Chapter 3 Exercise 3.2. Print and pin inside your Mathematics notebook.
Multiplying without the order check. If A is 2 × 3 and B is 2 × 3 , then AB does not exist. Writing a product that does not exist gets the entire 3-mark question struck out. Lose all 3 marks.
Computing A2 as element-squaring. A2 means A · A, not aij2. This is the single most common slip in Q16-Q19.
Assuming AB = BA . Matrix multiplication is non-commutative; even when both products exist they usually differ. Auto-deducts 2 marks if used silently.
Skipping the verdict line in property verification. CBSE awards a separate mark for the closing "Hence LHS = RHS." Computing both sides without stating the conclusion loses the easiest mark on the page.
Wrong identity matrix order. In A2 - 5A + 6I , I must be the same order as A. Subtracting a 3 × 3 identity from a 2 × 2 matrix is dimensionally invalid.
Returning only one matrix in "find X and Y" problems. Writing just X loses 2 of the 4 marks.
How to Study Class 12th Maths Exercise 3.2 in 6 Hours
Plan 6 hours across two sittings for a first pass; a revision sweep before the board paper takes 75 minutes.
Hour 1. Q1-Q3 (addition, scalar multiplication). Warm-ups; do not over-invest.
Hour 2. Q4-Q7 (matrix product). Write the order check as a labelled line.
Hour 3. Q8-Q10 (simultaneous matrix equations). The 4-mark goldmine of the exercise.
Hour 4. Q11-Q15 (distributive and associative properties). Two-column LHS/RHS layout.
Hour 5. Q16-Q19 (powers and matrix polynomial). Q19 type appears almost every year.
Hour 6. Q20-Q22 (non-commutativity and MCQs).
NCERT Solutions for Class 12 Maths Chapter 3 Matrices - All Exercises
Exercise 3.2 is the second of four exercises plus the Miscellaneous Exercise in Chapter 3. Wrap it before Ex 3.3 on transpose.
Exercise
Topic
Questions
Exercise 3.1
Order, types, equality of matrices
10
Exercise 3.2
Operations on matrices: addition, scalar product, matrix product, properties
22
Exercise 3.3
Transpose, symmetric and skew-symmetric matrices
12
Exercise 3.4
Elementary operations and inverse by elementary transformations
the resource: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Matrices Chapter
The Matrices chapter splits into 4 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
PDF Download Formats and Languages for the Matrices Chapter
The Matrices Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
Format
Best for
Approx. size
Normal-resolution PDF
Phone reading, quick revision between classes
2-3 MB
HD PDF
Print-ready, desk study, board hall photocopy
8-10 MB
Handwritten Notes PDF
Mirrors how a topper writes the chapter under Sunday-revision pace
5-7 MB
The matrices class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
NCERT-faithful: Every definition, theorem and exercise on the matrices class 12 ncert pdf matches the printed textbook line for line.
Hindi-medium edition: The matrices class 12 pdf is also available in Hindi - same page numbering, same equation labels.
Formula PDF separate: The matrices class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
Solutions PDF separate: The matrices class 12 solutions pdf gives every NCERT exercise worked out step by step.
State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this this chapter - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the matrices class 12 formulas pdf), and the full these notes on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Matrices Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Matrices chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
Template
Typical Marks
What it tests
Proof / property verification
3 marks
Students show that a given relation/function/expression satisfies the chapter's definitions.
One-step computation
2 marks
Substitution-based item: plug into a known formula and simplify.
Case-study scenario
4 marks
Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers.
Walking through one example of each template before the exam covers most of the predictable matrices class 12 important questions you will see on board day.
matrices class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
The matrices class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
For NCERT Exemplar practice, the matching matrices class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Matrices chapter across recent CBSE Class 12 Maths boards:
Year
Dominant Question Type
Approx. Marks
2024
Property verification + case-study item
5-6 marks
2023
Computation with proof + assertion-reason MCQ
5-6 marks
2022
Long-answer derivation + 2-mark substitution
5-7 marks
2021
Definition recall + property check
4-5 marks
2020
One-step computation + 3-mark proof
5 marks
The full matrices class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Matrices Notes Pair with NCERT Solutions and the Formula Sheet
The Matrices Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
Resource
Use it for
When
Matrices Notes (this page)
Theory, definitions, exam patterns
First pass, before practice
the PDF PDF
Step-by-step solved exercises
Second pass, during NCERT practice
this chapter formulas PDF
One-page identity recall
Third pass, alongside mock papers
Handwritten Notes PDF
Quick reading in topper's handwriting
Anytime, especially commute revision
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
The this chapter cover every back-of-chapter exercise plus the miscellaneous exercise.
The matrices class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
The these notes formulas reference sheet is the same A4 file students sometimes refer to as this Class 12 page all formulas - it lists every identity used in the chapter.
State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board the resource textbook PDF all share the same core definitions.
For class-first search phrasings - class 12 matrices solutions, class 12 matrices ncert solutions, ncert class 12 matrices solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
Reference
How it maps to the chapter notes
RD Sharma Class 12 Matrices
Question patterns overlap with NCERT at ~70%; an advanced supplement.
ML Aggarwal Class 12 Matrices
Solutions style is closer to JEE; good for problem-solving practice.
Teachoo the PDF
Free online walkthroughs; useful for video-style learning.
Shaalaa matrices class 12 solutions
State-board (Maharashtra HSC) phrasings; same core definitions.
Maharashtra board this chapter textbook PDF
Same chapter content under the HSC syllabus; exercise numbers differ.
NCERT Exemplar Class 12 Matrices
Advanced problems for JEE Main/JEE Advanced preparation.
NCERT Solutions for Class 12 Mathematics: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.
All NCERT Solutions for Matrices Ex 3.2 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 3 Matrices Ex 3.2 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Q 3.1
Let A=bmatrix 2 & 4 3 & 2 bmatrix, B=bmatrix 1 & 3 -2 & 5 bmatrix, C=bmatrix -2 & 5 3 & 4 bmatrix.
Find each of the following:
(i) A+B, (ii) A-B, (iii) 3A-C, (iv) AB, (v) BA.
Concept used. For matrices of the same order, addition and
subtraction are done entry-wise: (A± B)ij=aij± bij.
The scalar multiplekA multiplies every entry by k:
(kA)ij=k aij.
For multiplication of compatible matrices Am× nBn× p=Cm× p, the entry
cij=k=1naikbkj is the dot product of row i
of A with column j of B.
Row-by-column
(AB)ij= (row i of A) · (column j of B).
Always pair row of the left factor with column of the
right factor.
(iii) 3A-C. First scale A by 3:
3A=bmatrix 6 & 12 9 & 6 bmatrix.
Then subtract C:
3A-C=bmatrix 6-(-2) & 12-5 9-3 & 6-4 bmatrix
=bmatrix 8 & 7 6 & 2 bmatrix.
(iv) AB. Both are 2× 2; product is 2× 2.
Compute each entry as a row · column dot product:
(AB)11=(2,4)·(1,-2)=21+4·(-2)=2-8=-6.
(AB)12=(2,4)·(3,5)=23+45=6+20=26.
(AB)21=(3,2)·(1,-2)=3-4=-1.
(AB)22=(3,2)·(3,5)=9+10=19.
AB=bmatrix -6 & 26 -1 & 19 bmatrix.
(v) BA. Same way, but row i of B, column j of A:
(BA)11=(1,3)·(2,3)=2+9=11.
(BA)12=(1,3)·(4,2)=4+6=10.
(BA)21=(-2,5)·(2,3)=-4+15=11.
(BA)22=(-2,5)·(4,2)=-8+10=2.
BA=bmatrix 11 & 10 11 & 2 bmatrix.
Notice AB≠ BA, illustrating that matrix multiplication is
not commutative.
Strategic angle. Treat A+B, A-B, 3A-C as pure entry-wise
arithmetic (no row-column dance). For AB and BA, list each row of
the left matrix and each column of the right matrix as ordered pairs
and dot them.
Pre-list the data we need.
Rows of A: R1A=(2,4), R2A=(3,2).
Cols of A: C1A=(2,3), C2A=(4,2).
Rows of B: R1B=(1,3), R2B=(-2,5).
Cols of B: C1B=(1,-2), C2B=(3,5).
Structural observation. Look for algebraic identities before
computing: the second sum is a binomial-square mosaic; the trig sum is
the Pythagorean identity, four times.
(i) Routine entry sum; the -b and +b cancel out at (2,1).
(ii) Each entry has the shape p2± 2pq+q2, i.e.
(p± q)2. So the sum is the matrix of those squares.
bmatrix(a+b)2&(b+c)2 (a-c)2&(a-b)2bmatrix.
Concept used. For a product Am× nBn× p, each
entry (AB)ij=k=1naikbkj is the dot product of
row i of A with column j of B. The inner dimensions n
must match; the result has order m× p.
Strategic angle. For each pair, first check order compatibility,
then carry out the dot products. Watch the structure: (i) is the
classic ``rotation-times-rotation'' pattern that yields a scalar
multiple of the identity.
(i) The cross-terms cancel because of the sign pattern; only
the diagonal survives:
(a2+b2)I2.
(ii) Outer product of (1,2,3)T and (2,3,4). Each entry is
a product of one element from each. Yields the rank-1 matrix
bmatrix2&3&4 4&6&8 6&9&12bmatrix.
(iii) Pre-pair row entries with column entries and add:
(1,1)=-3,(1,2)=-4,(1,3)=1,(2,1)=8,(2,2)=13,(2,3)=9.
(iv) Same routine on 3× 3· 3× 3; nine entries
each a three-term dot product. Group by row to stay organised.
Each row of the result uses the same three rows of B paired
with one row of A.
(v), (vi) Same routine; track orders carefully. (v) gives a
3× 3, (vi) gives a 2× 2.
Why this matters. ``Order compatibility'' is the single
biggest pitfall in matrix multiplication problems. Always size-check
before multiplying.
Concept used. Addition and subtraction of matrices of the
same order is entry-wise. The associative property(A+B)+(-C)=A+(B+(-C)) for matrices is just associativity of addition
in R, applied component-wise.
Both results match entry-by-entry, so A+(B-C)=(A+B)-C.
A+B=bmatrix4&1&-1 9&2&7 3&-1&4bmatrix,
B-C=bmatrix-1&-2&0 4&-1&3 1&2&0bmatrix, and
A+(B-C)=(A+B)-C verified.
AP
Aanya Pillai
M.Sc Mathematics, IIT Bombay
Verified Expert
Quick reading. The verification is automatic because matrix
addition inherits associativity from R. Both computations
give the same nine numbers because they are evaluating the same
expression A+B-C.
Compute A+B row-by-row; result as above.
Compute B-C row-by-row; result as above.
For A+(B-C): add A and B-C entry-wise.
For (A+B)-C: subtract C from A+B entry-wise.
Both sequences end at bmatrix0&0&-3 9&-1&5 2&1&1bmatrix.
Why this matters. Associativity lets us drop the parentheses
in A+B-C; this is heavily used when collecting many matrix terms.
Both sides equal bmatrix0&0&-3 9&-1&5 2&1&1bmatrix.
Concept used. Scale each matrix by its scalar, then subtract
entry-wise. Common-denominator trick: multiplying A by 3 and B
by 5 clears the denominators 3 and 5 inside each matrix.
Compute 3A. Multiplying every entry of A by 3
clears the denominator 3:
3A=bmatrix 2 & 3 & 5 1 & 2 & 4 7 & 6 & 2 bmatrix.
Compute 5B. Multiplying every entry of B by 5
clears the denominator 5:
5B=bmatrix 2 & 3 & 5 1 & 2 & 4 7 & 6 & 2 bmatrix.
Subtract entry-wise.3A-5B has every entry equal to
0:
3A-5B=bmatrix 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 bmatrix=O.
3A-5B=O (the 3× 3 zero matrix).
TS
Tara Singh
M.Sc Mathematics, IIT Bombay
Verified Expert
Structural observation. The matrix B is exactly
35 of A (entry-by-entry). So 5B=3A, giving
3A-5B=0.
Test: Bij/Aij=2/52/3=25·32=35 for
(1,1); check (2,2): 2/52/3=35.
Pattern holds for every entry.
So B=35A, hence 5B=3A.
Therefore 3A-5B=3A-3A=O.
Why this matters. Spotting a scalar relationship between two
matrices instantly collapses a long arithmetic problem to a one-liner.
Strategic angle. Pull all unknowns to LHS in each scalar
equation. The diagonal entries decouple immediately because the
(1,1) and (2,2) equations don't involve other unknowns.
(1,1): 3x-x=4⇒ x=2.
(2,2): 3w-2w=3⇒ w=3.
(1,2): 3y-y=6+x=8⇒ y=4.
(2,1): 3z-z=-1+w=2⇒ z=1.
Why this matters. Decoupling –- solving the standalone
equations first –- is the standard tactic in 2x2 matrix systems with
mild dependencies.
(x,y,z,w)=(2,4,1,3).
Q 3.13
If F(x)=bmatrix cos x & -sin x & 0 sin x & cos x & 0 0 & 0 & 1 bmatrix, show that F(x) F(y)=F(x+y).
Concept used. Compute F(x)F(y) entry-by-entry and simplify
using the cosine/sine addition formulas
cos(x+y)=cos xcos y-sin xsin y and
sin(x+y)=sin xcos y+cos xsin y.
Write F(x) rows as
R1=(cos x,-sin x,0), R2=(sin x,cos x,0), R3=(0,0,1),
and F(y) columns as
C1=(cos y,sin y,0)T, C2=(-sin y,cos y,0)T, C3=(0,0,1)T.
Compute the nine entries of F(x)F(y). The last row and last
column are trivial (any pairing with the (0,0,1) axis gives
either 0 or 1):
(3,3): 1; (3,1),(3,2),(1,3),(2,3) all 0.
Collect into a matrix:
F(x)F(y)=bmatrixcos(x+y)&-sin(x+y)&0 sin(x+y)&cos(x+y)&0 0&0&1bmatrix
=F(x+y).
F(x) F(y)=F(x+y), proved by direct computation using the
sine and cosine addition identities.
KP
Kavya Pillai
Ph.D Mathematics, IIT Delhi
Verified Expert
Structural observation. Multiplying two block-diagonal
matrices is block-wise, so the bottom-right 1× 1 block stays
as 1· 1=1 and the top-left 2× 2 block multiplies on its
own. Treat it like a 2× 2 rotation problem.
Reduce to the 2× 2 block:
R(θ)=bmatrixcosθ&-sinθ sinθ&cosθbmatrix.
Compute R(x)R(y): each entry uses the relevant addition
formula. (1,1)=cos(x+y), (1,2)=-sin(x+y),
(2,1)=sin(x+y), (2,2)=cos(x+y).
Reassemble with the trivial 1× 1 block to get F(x+y).
Why this matters. ``Block-diagonal × block-diagonal =
block-diagonal,'' so a 3× 3 rotation problem reduces to a
2× 2 one. The same trick scales to n× n matrices with
block structure.
Concept used. For a square matrix A of order n,
A2=A· A. The identity matrix In satisfies kIn being
diagonal with k on the diagonal. The polynomial A2-5A+6I is
evaluated entry-wise after computing each term.
Compute A2=A· A. Rows of A:
R1=(2,0,1), R2=(2,1,3), R3=(1,-1,0).
Columns of A: C1=(2,2,1), C2=(0,1,-1), C3=(1,3,0).
Picture-first. Two products to compute (A2, A3), then a
plain entry-wise combination. The structure tells us A satisfies its
own characteristic polynomial λ3-6λ2+7λ+2.
A2 via 9 dot products (above).
A3=A· A2 via 9 more dot products (above).
Form A3-6A2+7A+2I entry-wise. The diagonal entries each
sum to 0 because A obeys its characteristic equation.
Verify off-diagonal (1,3) entry: 34-48+14+0=0 .
Why this matters. A matrix satisfies its own characteristic
polynomial (Cayley-Hamilton). This lets you compute powers of A as
linear combinations of I, A, A2, and is the standard route to
A-1 when known to exist.
O.
Q 3.17
If A=bmatrix 3 & -2 4 & -2 bmatrix and I=bmatrix 1 & 0 0 & 1 bmatrix, find k so that A2=kA-2I.
Concept used. Compute A2, match it against kA-2I
entry-wise, and solve for k.
I+A=(I-A)bmatrixcosα & -sinα sinα & cosαbmatrix, proved using
cosα=1-t21+t2, sinα=2t1+t2 with t=tan(α/2).
RV
Rahul Verma
Ph.D Pure Mathematics, IISc Bangalore
Verified Expert
Structural observation. Set t=tan(α/2); the algebra
reduces to verifying two identities, (1+t2)· stuff.
I+A, I-A are
bmatrix1&-t t&1bmatrix, bmatrix1&t -t&1bmatrix.
Multiply (I-A)R keeping t symbolic. The four entries
become 1+t21+t2=1 (diagonal) and
±t(1+t2)1+t2=± t (off-diagonal), matching I+A.
No need to expand fully; the 1+t2 factor cancels cleanly.
Why this matters. ``Half-angle'' identities are a coordinate
choice that simplify rotation matrices. The same parameter t
appears in stereographic projection and Cayley transform contexts.
Identity verified entry-by-entry.
Q 3.19
A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds, if the trust fund must obtain an annual total interest of (a) Rs. 1800, (b) Rs. 2000.
Concept used. Let x be the amount invested in the 5% bond
and 30000-x in the 7% bond. The total interest is the matrix product
[ x 30000-x ]bmatrix 5/100 7/100 bmatrix = (total interest).
(a) Target interest Rs. 1800. The matrix equation
is
[ x 30000-x ]bmatrix0.05 0.07bmatrix=1800.
Multiply out:
0.05x+0.07(30000-x)=1800.
Expand:
0.05x+2100-0.07x=1800.
Combine: -0.02x=1800-2100=-300.
x=-300-0.02=15000.
Then 30000-x=15000.
Allocation: Rs. 15,000 in 5% bond and
Rs. 15,000 in 7% bond.
(b) Target interest Rs. 2000. Same setup:
0.05x+0.07(30000-x)=2000. 0.05x+2100-0.07x=2000. -0.02x=-100⇒ x=-100-0.02=5000.
Then 30000-x=25000.
Allocation: Rs. 5,000 in 5% bond and
Rs. 25,000 in 7% bond.
Why this matters. Word problems on portfolio allocation,
inventory totals, and revenue all reduce to linear equations of this
type. Matrix notation lets you stack many such equations into AX=B.
(a) 15,000/15,000. (b) 5,000/25,000.
Q 3.20
The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books and 10 dozen economics books. Their selling prices are Rs. 80, Rs. 60 and Rs. 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
Concept used. Encode quantities (in copies, not dozens) as a
1× 3 row matrix and prices as a 3× 1 column matrix. The
1× 1 product is the total revenue.
Why this matters. The matrix way of writing
i qi pi generalises to many books in many shops, where the
quantity becomes a matrix and the price becomes a column.
Rs. 20,160.
Q 3.21
Assume X,Y,Z,W,P are matrices of order 2× n, 3× k, 2× p, n× 3, p× k, respectively. The restriction on n,k and p so that PY+WY will be defined are:
(A) k=3, p=n (B) k is arbitrary, p=2 (C) p is arbitrary, k=3 (D) k=2, p=3.
Concept used. For AB to be defined, the number of columns
of A must equal the number of rows of B. For A+B to be defined,
A and B must have the same order.
P is p× k, Y is 3× k. For PY to be defined,
columns of P (=k) must equal rows of Y (=3), so
k=3. Then PY has order p× k=p× 3.
W is n× 3, Y is 3× k. For WY, columns of
W (=3) match rows of Y (=3) automatically. Then WY
has order n× k=n× 3 (since k=3).
For PY+WY to be defined, the orders must match:
p× 3 = n× 3, so p=n.
Why this matters. Quick order-checks save tons of arithmetic
in exam settings.
(A).
Q 3.22
Assume X,Y,Z,W,P are as in Q21. If n=p, then the order of the matrix 7X-5Z is:
(A) p× 2 (B) 2× n (C) n× 3 (D) p× n.
Concept used.kA has the same order as A. A-B requires
A and B to have the same order, and the result has that same
order.
X has order 2× n; 7X has order 2× n.
Z has order 2× p; 5Z has order 2× p.
For 7X-5Z to be defined, 2× n = 2× p, so n=p
(given).
Common order: 2× n. Among the choices, that is (B).
Correct answer: (B) 2× n.
AS
Aditya Singh
M.Tech CS, IIT Madras
Verified Expert
Quick reading. Scalars preserve order; subtraction needs
matching orders; result keeps that order.
7X is 2× n, 5Z is 2× p=2× n (since n=p).
Their difference is 2× n.
Option (B).
Why this matters. Tracking ``order'' is the cheapest sanity
check; nail it before any arithmetic.
(B) 2× n.
How to Use the Matrices Notes Page Most Effectively
The recommended study plan for these notes chapter splits across three sittings. The table below outlines what to do in each:
Sitting
Duration
What to do
Sitting 1: Theory
~90 minutes
Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page.
Sitting 2: Solved Examples
~90 minutes
Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use these notes PDF if stuck.
Sitting 3: Exercises
~90 minutes
Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification.
For students preparing for both CBSE board and JEE Main:
60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
The matrices class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Matrices Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Exercise 3.2 of Class 12 Maths Chapter 3 Matrices?
Ans. Exercise 3.2 of NCERT Class 12 Maths Chapter 3 Matrices contains 22 questions in total. The set covers addition, scalar multiplication, matrix product, distributive and associative properties, powers of a square matrix, simultaneous matrix equations in X and Y, and demonstrations that matrix multiplication is non-commutative.
Ques. Where can I download the the PDF for free?
Ans. You can download the Class 12 Maths Chapter 3 Matrices Exercise 3.2 NCERT Solutions PDF directly from the this chapter. Both the Normal and HD versions are free, and a handwritten-style version is also available. these notes is solved by Collegedunia subject experts as per the 2026-27 NCERT.
Ques. Is Class 12 Maths Exercise 3.2 part of the 2026-27 CBSE syllabus?
Ans. Yes. Matrices remains a full chapter in the 2026-27 NCERT Class 12 Maths syllabus and Exercise 3.2 is intact with all 22 questions. The new edition keeps every operation, property, and proof in Exercise 3.2 unchanged from the previous print.
Ques. Which questions of Exercise 3.2 are most important for the CBSE Class 12 board exam?
Ans. Questions Q8, Q9, Q10 (simultaneous matrix equations), Q11-Q15 (property verifications), and Q19 (matrix polynomial of the form A2 - 5A + 6I ) carry the highest CBSE weight. Q19-type appears almost every year in CBSE Class 12 or JEE Main with minor numerical changes.
Ques. How do you check if the product AB of two matrices exists?
Ans. The product AB exists only when the number of columns of A equals the number of rows of B.
If A is of order m × n and B is of order n × p, then AB is defined and is of order m × p. If the inner dimensions do not match, the product is undefined and you should state this explicitly in your CBSE answer.
Ques. Is matrix multiplication commutative in Class 12 Maths Exercise 3.2?
Ans. No. Matrix multiplication is in general non-commutative, meaning AB ≠ BA for most matrices A and B. In some cases BA may not even exist while AB does. Exercise 3.2 includes specific questions (Q20, Q21, Q22) that ask students to demonstrate this non-commutativity using NCERT-style 2x2 and 2x3 matrices.
Ques. How long should it take to complete Class 12th Maths Chapter 3 Exercise 3.2?
Ans. Plan for 5 to 7 hours across two or three sittings if you are seeing Exercise 3.2 for the first time. A revision pass before the CBSE Class 12 board paper takes roughly 75 to 90 minutes once you have already solved the 22 questions once.
Ques. What is the difference between A2 and squaring each entry of matrix A?
Ans. A2 means the matrix product A · A, computed using the row-by-column rule. It is NOT the matrix obtained by squaring each entry aij.
For example, if A = bmatrix 1 & 2 3 & 4 bmatrix , then A2 = bmatrix 7 & 10 15 & 22 bmatrix , not bmatrix 1 & 4 9 & 16 bmatrix . Confusing the two is the single most common error in Class 12 Maths Exercise 3.2.
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