Question

Give examples of two matrices A and B of the same order for which AB = O but BA ≠ O.

Answer 1

O stands for Null Matrix.

Here are the two matrices of SAME order for which the given condition AB = O but BA ≠ O satisfies.

Both of these Matrices are of Order 2 by 2.

$A = \left[\begin{array}{cc}5&4\\5&4\\\end{array}\right]\\ B = \left[\begin{array}{cc}4&4\\-5&-5\\\end{array}\right]$

$AB = \left[\begin{array}{cc}5&4\\5&4\\\end{array}\right]\left[\begin{array}{cc}4&4\\-5&-5\\\end{array}\right] \\AB = \left[\begin{array}{cc}5(4)+4(-5)&5(4)+4(-5)\\5(4)+4(-5)&5(4)+4(-5)\\\end{array}\right]\\AB = \left[\begin{array}{cc}0&0\\0&0\\\end{array}\right]$

so

AB = O

Also

$BA = \left[\begin{array}{cc}4&4\\-5&-5\\\end{array}\right] \left[\begin{array}{cc}5&4\\5&4\\\end{array}\right]\\BA = \left[\begin{array}{cc}4(5)+4(5)&4(4)+4(4)\\-5(5)+(-5)(5)&-5(4)+(-5)(4)\\\end{array}\right]\\BA = \left[\begin{array}{cc}40&32\\-50&-40\\\end{array}\right]\\BA\neq O$