Example 24 - Chapter 3 Class 12 Matrices
Last updated at April 16, 2024 by Teachoo
Proof using property of transpose
Proof using property of transpose
Last updated at April 16, 2024 by Teachoo
Example 24 If A and B are symmetric matrixes of the same order, then show that AB is symmetric if and only if A and B commute, that is AB = BA. Given A & B are symmetric matrix i.e. A’ = A B’ = B We need to show AB is symmetric if and only if A & B commute (i.e. AB = BA) i.e. we need to show If AB is symmetric, then A & B commute (i.e. AB = BA) and If A & B commute (i.e. AB = BA), then AB is symmetric Proving Forward part If AB is symmetric then A & B commute Given AB is symmetric i.e. (AB)’ = AB B’A’ = AB BA = AB Hence A & B commute. Hence proved Proving backward part If A & B commute, then AB is symmetric Given A & B commute i.e. AB = BA We need to show AB is symmetric i.e. we need to show (AB)’ = AB Taking (AB)’ = B’A’ = BA = AB So, (AB)’ = AB Hence, AB is symmetric Hence proved Therefore, AB is symmetric if and only if A and B commute, i.e. AB = BA.