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Scalar Matrix is a kind of diagonal matrix with similar elements on its diagonals. If all the elements of a scalar matrix have a value of 1, then it is identified as an identity matrix. A scalar matrix has an n × n order. In this article, we will learn more about scalar matrix, their properties, formula, and various operations.
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Key Terms: Scalar Matrix, diagonal matrix, identity matrix, squre matrix, scalar multiplication, formula, order
Matrix
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A matrix is a rectangular array or a table in which numbers or components are arranged in rows and columns. Matrices are the plural version of the matrix. Matrices can contain as many columns and rows as they want. Matrix addition, scalar multiplication, multiplication, transposition, and other operations can be done on them.
Read more: Types of Matrices
When conducting these matrix operations, certain rules must be followed, such as matrices can only be added or subtracted if they have the same number of rows and columns, and they can only be multiplied if the columns in the first matrix and rows in the second one are precisely the same.
The video below explains this:
Matrices Detailed Video Explanation:
Scalar Matrix
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A diagonal matrix is known as a scalar matrix. The diagonal values of these matrices are equal or identical. When all of the values of a scalar matrix are equal to 1, it is called an identity matrix.
A square matrix A = [aij]n × n is called a scalar matrix if it meets the following criteria:
- aij = 0, when i ≠ j
- aij = k, when i = j, for some constant k
The order for the scalar matrix is n × n. As a result, there is the same number of rows and columns in the scalar matrix. Hence, a scalar matrix is also a square matrix.
Example:
A scalar matrix is a square matrix with all the off-diagonal elements equal to zero and all the on-diagonal elements having the same value. A scalar matrix is a multiple of an identity matrix that contains any scalar quantity.
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| Operations in Matrices | Symmetric Matrices | Transpose of Matrix |
Scalar Multiplication
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Real numbers are referred to as scalars while working with matrices. The product of a real number with a matrix is known as scalar multiplication. Each entry in the matrix is multiplied by the supplied scalar in scalar multiplication.
\(A = \begin{bmatrix} a&0&0\\ 0&a&0\\ 0&0&a \end{bmatrix}\)
The primary diagonal elements of the above matrix are all equal to the same numeric value of 'a,' and all other matrix components are equal to zero. The scalar matrix is derived from an identity matrix, with the scalar matrix being the product of the identity matrix and a constant value. The scalar matrix is obtained by multiplying the constant 'a' with an identity matrix, and the order of this matrix is 3 × 3.
Example - Given that A = \(\begin{bmatrix} 10&5\\ 5&10 \end{bmatrix}\), find 2A.
Solution: To find 2A, simply multiply each matrix entry by 2:
2A = \(2\begin{bmatrix} 10&5\\ 5&10 \end{bmatrix}\)
2A = \(\begin{bmatrix} 2\times10&2\times5\\ 2\times5&2\times10 \end{bmatrix}\)
= \(\begin{bmatrix} 20&10\\ 10&20 \end{bmatrix}\)
Formula for Scalar Matrix
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Constant × Identity Matrix = Scalar Matrix
\(k \times \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix} = \begin{bmatrix} k&0&0\\ 0&k&0\\ 0&0&k \end{bmatrix}\)
Scalar Matrix Properties
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The scalar matrix has the following properties:
- All scalar matrices are symmetric matrices as well.
- A scalar matrix is both an upper triangular matrix and a lower triangular matrix.
- The product of an identity matrix with a scalar integer yield a scalar matrix.
- The zero matrix is also a scalar matrix.
Along with the scalar matrix, there are various other matrices such as null matrix, column matrix, row matrix, etc.
Read more: Invertible Matrices
Multiplication of Scalar Matrix
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- (k + l)A = kA + lA
Let us take the LHS
= (k + l)A
= (k + l)[aij]
= [(k + l) aij] + [k.aij] + [l.aij]
= k[aij] + l[aij]
= kA + lA
- k(A+B) = kA + kB
Taking the LHS
= k(A + B)
= k([aij] + [bij])
= k[aij + bij]
= [k(aij + bij)]
= [(k.aij) + (k.bij)]
= [k.aij] + [k.bij]
= k[aij] + k[bij]
= kA + kB
Terms Related to Scalar Matrix
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Identity Matrix
The identity matrix is a square matrix with a multiplicative identity. All the diagonal members of the identity matrix are 1, and all other elements are equal to zero. The identity matrix has several uses in matrix multiplication and determining the inverse of a matrix. The identity matrix produces a scalar matrix when multiplied by a constant value.
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Diagonal Matrix
The diagonal matrix is also a square matrix with components of varying values along the primary diagonal and all other members equal to zero. Furthermore, if all of the diagonal elements of the diagonal matrix are made equal, the matrix is referred to as a scalar matrix.
Also Read:
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| Matrices Handwritten Notes | Class 12 NCERT Maths Chapter 3 Matrices Solutions | Matrices Important Questions |
| Symmetric and Skew Symmetric Matrices | Orthogonal Matrix | Determinants |
Things to Remember
- Scalar Matrix is a kind of diagonal matrix with similar elements on its diagonals.
- A square matrix is a scalar matrix.
- All off-diagonal elements are equal to zero.
- The on-diagonal elements are all the same.
- It's formed by multiplying the identity matrix with a scalar.
Sample Questions
Ques. What is a scalar matrix? What is its order? (2 Marks)
Ans. Scalar Matrix is a diagonal matrix with similar elements on its diagonals. A scalar matrix has an n × n order
Ques. When can a scalar matrix be regarded as an identity matrix? (2 Marks)
Ans. If all the elements of a scalar matrix have a value of 1, then it is identified as an identity matrix.
\(I = \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}\)
Ques. Calculate the determinant of a Scalar Matrix A = \(\begin{bmatrix} 9&0&0\\ 0&9&0\\ 0&0&9 \end{bmatrix}\) (3 Marks)
Ans. Given Matrix,
A = \(\begin{bmatrix} 9&0&0\\ 0&9&0\\ 0&0&9 \end{bmatrix}\)
It can be also written as:
A = \(9 \times \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}\)
A = 9 × I
|A| = 9 × |I| = 9 × 1 = 9
Hence the determinant of the given scalar matrix is 9.
Ques. If A = \(\begin{bmatrix} 2&3\\ 1&2 \end{bmatrix}\), B = \(\begin{bmatrix} 1&3&2\\ 4&3&1 \end{bmatrix}\), C = \(\begin{bmatrix} 1\\ 2 \end{bmatrix}\) , D = \(\begin{bmatrix} 5&7&9\\ 4&6&8 \end{bmatrix}\)
Then which of the following sums is defined: A + B, B + C, C + D, and B + D? (1 Mark)
Ans. As only the matrices of the same order can only be added, therefore only B + D is defined.
Ques. If A and B are square matrices of the same order, then (A + B) (A – B) is equal to: (1 Mark)
(a) A2 - B2
(b) A2 - BA - AB - B2
(c) A2 - B2 + BA - AB
(d) A2 - BA + B2 + AB
Ans. The correct answer is (c)
(A+B) (A-B) = A(A-B) +B(A-B)
= A2 - B2 + BA - AB
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