Question:

Let A and B be any two 3 × 3 symmetric and skew symmetric matrices, respectively. Then Which of the following is NOT true?

Updated On: Apr 12, 2025
  • A4 – B4 is a symmetric matrix
  • AB – BA is a symmetric matrix
  • B5 – A5 is a skew-symmetric matrix
  • AB + BA is a skew-symmetric matrix
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The Correct Option is C

Solution and Explanation

Given: AT = A, BT = -B 
From option 1: Let C = A4 - B4 
CT = (A4 - B4) = (A4)T - (B4)T = A4 - B4 = C 
From option 2: Let C = AB - BA 
CT = (AB - BA)T = (AB)T - (BA)T 
= BTAT - ATBT = -BA + AB = C 
From option 3: Let C = B5 - A5 
CT = (B5 - A5)T = (B5)T - (A5)T = -B5 - A5 
From option 4: Let C = AB + BA 
CT = (AB + BA)T = (AB)T + (BA)T = -BA - AB = -C 
∴ Option 3 is not true.
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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.