MATRICES — Maths STD 12 Science — Question

Given that,

A and B are symmetric matrices, so

⇒ A^T = A and B^T = B

Now,

$\big(\text{AB}\big)^\text{T}=\text{B}^\text{T}\times\text{A}^\text{T}$ $\big\{\text{since, (AB)}^\text{T}=\text{B}^\text{T}\text{A}^\text{T}\big\}$

$\big(\text{AB}\big)^\text{T}=\text{BA}\ \dots(\text{i})$ $\big\{\text{since, B}^\text{T}=\text{B},\text{A}^\text{T}=\text{A}\big\}$

For AB to be symmetric matrix

(AB)^T = AB

From equation (i) and (ii),

AB = BA

So,

For AB to be symmetric matrix we must have AB = BA.