CBSE Class 12 Mathematics Notes Chapter 4 Determinants

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Determinants is a scalar value that is calculated using the elements of a square matrix. It will return a single number as output.

In simpler terms, every square matrix of order n that can relate to a number is known as the determinant of a square matrix.
It can be represented as follows: det(A), det A, or |A|.
The concept helps in solving linear equations.
It is considered a scaling factor that is used for transforming matrices.
Determinants are used to determine the adjoint and inverse of the matrix.
It is used to determine the uniqueness of solutions in the matrix.
The main difference between determinants and matrices is that a matrix refers to an array of numbers, and a determinant refers to the special number of a square matrix.

According to the CBSE Syllabus 2023-24, the chapter on Determinants comes under Unit 2 of Algebra. NCERT Class 12 Mathematics Unit Algebra holds a weightage of around 10 marks and includes matrices and determinants topics.

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Class 12 Mathematics Chapter 4 Notes – Determinants

Types of Determinants

Determinants are divided into three categories, which are as follows:

First Order Determinant

First-order determinants are a type of determinant where the matrix is of order one. Suppose [a]=A, then in such a case, the determinant of A will be equal to ‘a’.

A = [a]1×1

Example of First Order Determinant

Example: A = [2]1×1

Second Order Determinant

Second Order Determinant is a type of determinant where the matrix is of order two.

It is calculated by multiplying the diagonally opposite elements and finding the difference between these two products.

det \(\begin{pmatrix}a & b \\[0.3em]c& d\\[0.3em] \end{pmatrix} \) = \(\begin{bmatrix}a& b\\[0.3em]c& d\\[0.3em] \end{bmatrix} \) = ad – bc

Example of Second Order Determinant

Example: The example of determinants of 2 x 2 matrix is as follows:

det \(\begin{bmatrix}3 & 5\\[0.3em]4& 6\\[0.3em] \end{bmatrix} \) = (3)(6) – (4)(5) = 18 – 20 = 2

Third Order Determinant

Third Order Determinant is a type of determinant where the matrix is of order three.

In this type of determinant, first add the product of the diagonally opposite elements.
Next, subtract the sum of elements perpendicular to the line segment.

\(\begin{bmatrix}a & b & c\\[0.3em]d&e&f \\[0.3em]g&h&i \\[0.3em] \end{bmatrix} \) = aei + bfg + cdh – cg – bdi – afh

Example of Third Order Determinant

Example: The example of determinants of 3 x 3 matrix is as follows:

det \(\begin{bmatrix}2 & 4 & 1\\[0.3em]3&4&1 \\[0.3em]5&1&3 \\[0.3em] \end{bmatrix} \) = 2 (12-1) – 4(9 – 5) + 1 (3 – 20)

= 22 – 16 – 17

= – 11

Minors of a Determinant

Minors of a determinant are determined by deleting the row and column in which that element lies.

It involves the deletion of the ‘ith’ row and the ‘jth’ column in which the aij element lies.

Example of Minor Determinant

Example: Consider a 3 x 3 determinant

∣a₁₁ a₁₂ a₁₃∣

∣a₂₁ a₂₂ a₂₃∣

∣a₃₁ a₃₂ a₃₃∣

Then minor of a₁₂ is given as: ∣a₂₁ a₂₃∣

∣a₃₁ a₃₃∣

Cofactors of Determinant

Cofactors of a determinant are related to the minor of a determinant. It is obtained by multiplying the minor of an element with -1 by the exponent(power) of the sum of the ith row and jth row.

It is denoted by A_ij= (-1)^i+jM_ij.

Example of Cofactor Determinant

Example: Consider a 3 x 3 determinant

∣a₁₁ a₁₂ a₁₃∣

∣a₂₁ a₂₂ a₂₃∣

∣a₃₁ a₃₂ a₃₃∣

Then cofactor of a₁₂ is given as: (-1)^{1 + 1} ∣a₂₁ a₂₃∣

∣a₃₁ a₃₃∣

Application of Determinants in finding Area of triangle

The area of the triangle using a determinant is calculated in the field of coordinate geometry. Generally, it is calculated using half the product of the base and altitude of the triangle.

The area of the triangle using the determinant is calculated when only vertices are given, and height is unknown.
Consider the triangle ABC with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃).
Area of the triangle can be calculated as (1/2) [x₁ (y₂ - y₃) + x₂ (y₃ - y₁) + x₃ (y₁ - y₂)].
In the case of a determinant, it is a scalar quantity that has positive and negative values.
If the area is negative, we will consider the absolute value of the determinant.

Area of the Triangle using determinant is given as: ½ x ∣a₁₁ a₁₂ 1∣

∣a₂₁ a₂₂ 1∣

∣a₃₁ a₃₂ 1∣

Area of Δ = \(\frac{1}{2}\)\(\begin{bmatrix}x_1& y_1 & 1\\[0.3em]x_2&y_2 &1 \\[0.3em]x_3&y_3&1 \\[0.3em] \end{bmatrix} \)

Area of Triangle using Determinants

Adjoint of Square Matrix

Adjoint of a square matrix is defined as the transpose of the cofactors of all the elements of the required matrix.

It is also known as an adjugate of a matrix.
Adjoint of a square matrix is denoted by adj A.
Transpose of a matrix involves the replacement of rows with columns and vice-versa.

Example of Adjoint of Square Matrix

Example: Determine the adjoint of the given square matrix.

C = ∣9 6∣

∣7 2∣

Then adjoint of a matrix is given as: C = ∣9 7∣

∣6 2∣

[C] = \(\begin{bmatrix}-61&11&-34\\[0.3em]-17&-41 &-26\\[0.3em]-53&-29&-2 \\[0.3em] \end{bmatrix} \), adj [A] = \(\begin{bmatrix}-61&-17&-53\\[0.3em]11&-41 &-29\\[0.3em]-34&-26&-2 \\[0.3em] \end{bmatrix} \)

Adjoint of Matrix

Inverse of a Square Matrix

The inverse of a square matrix is defined as the division of the adjoint of a matrix using the determinant of the matrix.

It is denoted by A^-1.
When the inverse matrix is multiplied by the given matrix, then it will give the identity matrix.
It can be mathematically represented as: AA^-1 = A^-1A = I, where I is the Identity matrix

A^-1 = \(\frac{1}{|A|} adj A\)

Inverse of a Square Matrix

Example of Inverse of a Square Matrix

A = \(\begin{bmatrix}a & b \\[0.3em]c & d \\[0.3em] \end{bmatrix} \)

The inverse of a matrix id found using the following formula:

A^-1= \(\begin{bmatrix}a & b \\[0.3em]c & d \\[0.3em] \end{bmatrix}^{-1}\)

A^-1= \(\frac{1}{ad - bc}\begin{bmatrix}d & -b \\[0.3em]-c & a \\[0.3em] \end{bmatrix} \)

Inverse of a Square Matrix

Solving Linear Equations Using Matrix

Solving Linear Equations can be done using the matrix method and row reduction method, also known as the Gaussian elimination method.

Consider the linear equations:

a₁x + a₂y + a₃z = d₁

b₁x + b₂y + b₃z = d₂

c₁x + c₂y + c₃z = d₃

First Method

In the matrix method, first, write all variables in a particular order. Next, write the variables, their coefficients and constants on their respective sides.

Solving linear equations using a matrix consists of forming two new matrices.
Let A be the first matrix which represents all variables.
B is the other matrix, which represents all constants

B is the other matrix, which represents all constants

AX=B…..(i)
Where, A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Where, A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Now multiply (i) by A⁻¹ and we will get:
A⁻¹AX=A⁻¹B
I.X=A⁻¹B
X=A⁻¹B

Example of Solving Linear Equation using Matrix

Example of Solving Linear Equation using Matrix

Second Method

The second method to solve linear equations involves the use of a Gaussian method or row reduction method.

First, the augmented matrix for linear equations.
Next, the elementary method is used so that all the elements below the main diagonal are equivalent to zero.
In case zero is obtained on the main diagonal, then use the row method to get a non-zero element.
Lastly, use the substitution method to solve the equations.

Consistent and Inconsistent Equations

An equation is said to be consistent if it has one or more solutions.
Similarly, if a set of equations has no solutions, then it is called inconsistent equations.

Conditions for Consistency of a Linear Equations

If |A| ≠ 0, then in such conditions, linear equations are said to be consistent and give a unique solution, which is given by X = A-1 B
If |A| = 0 and (Adj A) B ≠ 0, then in such conditions, linear equations are inconsistent.
If |A| = 0 and (Adj A) B = 0, then in such conditions, linear equations are consistent and give infinitely many solutions.

Conditions for Consistency of a Linear Equations

There are Some important List Of Top Mathematics Questions On Determinants Asked In CBSE CLASS XII

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CBSE Class 12 Mathematics Notes Chapter 4 Determinants

Class 12 Mathematics Chapter 4 Notes – Determinants

CBSE CLASS XII Related Questions

1.
If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that \[ QR = \frac{3}{2} QP. \]

2.
Let $|\vec{a}| = 5 \text{ and } -2 \leq \lambda \leq 1$. Then, the range of $|\lambda \vec{a}|$ is:

3.
If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:

4.
Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.

5.
Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{5 \sin x + 3 \cos x}{\sin x + \cos x} \, dx \]

6.
Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]

Similar Mathematics Concepts

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CBSE Class 12 Mathematics Notes Chapter 4 Determinants

Class 12 Mathematics Chapter 4 Notes – Determinants

CBSE CLASS XII Related Questions

1.If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that \[ QR = \frac{3}{2} QP. \]

2.Let $|\vec{a}| = 5 \text{ and } -2 \leq \lambda \leq 1$. Then, the range of $|\lambda \vec{a}|$ is:

3.If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:

4.Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.

5. Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{5 \sin x + 3 \cos x}{\sin x + \cos x} \, dx \]

6. Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that \[ QR = \frac{3}{2} QP. \]

2.
Let $|\vec{a}| = 5 \text{ and } -2 \leq \lambda \leq 1$. Then, the range of $|\lambda \vec{a}|$ is:

3.
If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:

4.
Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.

5.
Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{5 \sin x + 3 \cos x}{\sin x + \cos x} \, dx \]

6.
Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]