Question:
Inverse of a diagonal matrix (if it exists) is a
Inverse of a diagonal matrix (if it exists) is a
Updated On: Jul 6, 2022
- skew symmetric matrix
- non-invertible matrix
- diagonal matrix
- none of these
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The Correct Option is C
Solution and Explanation
Let A= diag $(d_1, d_2, d_3, ... , d_n)$.
As A is invertible, therefore, det $A \neq 0$
$\Rightarrow \, d_1 \, d_2 \, d_3 ... d_n \neq 0$
$\Rightarrow \, d_i \neq 0$ for $i = 1, 2, 3, ... , n,$
Here, cofactor of each non-diagonal entry is 0 and coafctor of $a_{ij}$
$ = (-1)^{i + i} $ det (diag ${d_1, d_2 ,......, d_{i-1} , d_{i +1}, ...., d_n)}$
$ = d_1 d_2 ...... d_{i-1} d_{i+1} .... d_n$
$ = \frac{1}{d_i} (d_1 d_2 ... d_{i-1} d_i d_{i+1} .... d_n) = \frac{|A|}{d_i}$
$\therefore \, A^{-1} = \frac{1}{|A|} \left(\text{adj} A\right) = \text{diag} \left(\frac{1}{d_{1} }, \frac{1}{d_{2} },...., \frac{1}{d_{n}}\right) $ which is a diagonal matrix.
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.