Question

For any square matrix A, show that AA' is symmetric.

Answer 1

let A be any matrix.

.$\text { Also let } \mathrm{B}=\mathrm{AA}^{\mathrm{'}}\\\text { Now } \mathrm{B}^{\mathrm{'}}=\left(\mathrm{AA}^{\mathrm{'}}\right)^{\mathrm{'}}=\left(\mathrm{A}^{\mathrm{'}}\right)^{\mathrm{'}} \mathrm{A}^{\mathrm{'}}=\mathrm{AA}^{\mathrm{'}}=\mathrm{B} .\left[\text { Since }\left(\mathrm{A}^{\mathrm{'}}\right)^{\mathrm{'}}=\mathrm{A}\right]$

$\text { So } \mathrm{AA}^{\mathrm{'}} \text { is a symmetric matrix. }$