Question:

If $ A$ and $B$ are matrices and $B = ABA^{-1}$ then the value of $(A + B) (A - B)$ is

Updated On: Apr 28, 2024
  • $A^2 + B^2$
  • $A^2 - B^2$
  • $A + B $
  • $A - B $
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The Correct Option is B

Solution and Explanation

$B = ABA^{-1}$ (Given) But $ B = BAA^{-A}$ $ \therefore \, \, ABA^{-1} = BAA^{-1} \, \, \Rightarrow \, \, AB = BA$ Now $(A + B) (A - B) = A^2 - AB + BA - B^2$ $= A^2 - AB + AB - B^2 \, \, \, [ \therefore \, \, AB = BA]$ $ = A^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:

  1. 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.
  2. Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
  3. 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. 
  4. 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. 
  5. Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.