Problem 214 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A\) and \(B\) be symmetric matrices of the same order, then show that i. \(A+B\) is a symmetric matrix ii. \(A B-B A\) is a skew-symmetric matrix iii. \(A B+B A\) is a symmetric matrix

Step-by-Step Solution

Answer

The sum \(A+B\) and the sum \(AB+BA\) of two symmetric matrices A and B are symmetric matrices, and the difference \(AB-BA\) is a skew-symmetric matrix.

Step 1: Proving A+B is symmetric

To prove that \(A+B\) is symmetric, we need to check whether \( (A+B)^T = A+B\). According to the properties of matrix transposition, \( (A+B)^T = A^T + B^T \). Since A and B are symmetric matrices, \( A^T=A\) and \( B^T=B \), thus \( (A+B)^T = A + B\). Therefore, \(A+B\) is a symmetric matrix.

Step 2: Proving AB-BA is skew-symmetric

To prove that \(AB-BA\) is skew-symmetric, we need to check if \((AB - BA)^T = -(AB - BA)\). According to the properties of matrix transposition, \( (AB - BA)^T = B^TA^T - A^TB^T \). But, since A and B are symmetric matrices, \( A^T = A\) and \( B^T = B \), so \( (AB - BA)^T = BA - AB \). Thus, \( (AB - BA)^T = -(AB - BA) \), which makes \( AB - BA\) a skew-symmetric matrix.

Step 3: Proving AB+BA is symmetric

To prove that \(AB + BA\) is symmetric, we need to check if \((AB + BA)^T = AB + BA\). According to the properties of matrix transposition, \( (AB + BA)^T = B^TA^T + A^TB^T \). But, since A and B are symmetric matrices, \( A^T = A\) and \( B^T = B \), so \( (AB + BA)^T = BA + AB \). However, in terms of addition of matrices, \( AB + BA = BA + AB \), Thus, \((AB + BA)^T = AB + BA \), which makes \( AB + BA\) a symmetric matrix.