Question:
If $A = \begin{bmatrix}a&0&0\\ 0&a&0\\ 0&0&a\end{bmatrix}$, then det(adj A) is
If $A = \begin{bmatrix}a&0&0\\ 0&a&0\\ 0&0&a\end{bmatrix}$, then det(adj A) is
Updated On: June 02, 2025
- $a^{27}$
- $a^{9}$
- $a^{6}$
- $a^{2}$
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The Correct Option is C
Solution and Explanation
Since $A = \begin{bmatrix}a&0&0\\ 0&a&0\\ 0&0&a\end{bmatrix}$
$ \Rightarrow \:\:\:\: |A| = a.a.a =a^3$
Using formula $[ {adj} \, Al = |A|^{n-1}$, we get det $( {adj} \, A)= (a^3)^{3-1} = (a^3)^2 = a^6$
$ \Rightarrow \:\:\:\: |A| = a.a.a =a^3$
Using formula $[ {adj} \, Al = |A|^{n-1}$, we get det $( {adj} \, A)= (a^3)^{3-1} = (a^3)^2 = a^6$
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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.