Question:
If A and B are square matrices of size n ? n such that $A^2 - B^2 = (A- B)(A+ B)$ , then which of the following will be always true?
If A and B are square matrices of size n ? n such that $A^2 - B^2 = (A- B)(A+ B)$ , then which of the following will be always true?
Updated On: Jul 6, 2022
- A = B
- AB = BA
- either of A or B is a zero matrix
- either of A or B is identity matrix
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The Correct Option is B
Solution and Explanation
$A^{2} - B^{2} = \left(A -B\right)\left(A+B\right) $
$A^{2} - B^{2} = A^{2} + AB - BA - B^{2}$
$ \Rightarrow AB = BA $
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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.