Question:
If $A$ and $B$ are square matrices of the same order such that $(A + B) (A -B) = A^2-B^2$ then $(ABA^{-1} )^2=$
If $A$ and $B$ are square matrices of the same order such that $(A + B) (A -B) = A^2-B^2$ then $(ABA^{-1} )^2=$
Updated On: May 18, 2024
- $A^2$
- $B^2$
- $I$
- $A^2B^2$
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The Correct Option is B
Solution and Explanation
Given, $(A+B)(A-B)=A^{2}-B^{2}$
$\Rightarrow A^{2}-A B+B A-B^{2}=A^{2}-B^{2}$
$\Rightarrow A B=B A$
Now, $\left(A B A^{-1}\right)^{2}=\left(B A A^{-1}\right)^{2}=B^{2}$
$\Rightarrow A^{2}-A B+B A-B^{2}=A^{2}-B^{2}$
$\Rightarrow A B=B A$
Now, $\left(A B A^{-1}\right)^{2}=\left(B A A^{-1}\right)^{2}=B^{2}$
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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.