Show that AB ≠ BA in the following matrices: and .

Engineering

Mathematics

Algebra of Matrices

Multiplication of Matrices

Special Type of Square Matrices

Question

Show that AB ≠ BA in the following matrices:
$A = [\begin{array}{ccc} 1 & 3 & - 1 \\ 2 & - 1 & - 1 \\ 3 & 0 & - 1 \end{array}]$ and $B = [\begin{array}{ccc} - 2 & 3 & - 1 \\ - 1 & 2 & - 1 \\ - 6 & 9 & - 4 \end{array}]$ .

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$A = [\begin{array}{ccc} 1 & 3 & - 1 \\ 2 & - 1 & - 1 \\ 3 & 0 & - 1 \end{array}] B = [\begin{array}{ccc} - 2 & 3 & - 1 \\ - 1 & 2 & - 1 \\ 6 & 9 & - 4 \end{array}]$

$AB = [\begin{array}{ccc} - 11 & 0 & 0 \\ - 9 & - 5 & 3 \\ - 12 & 0 & 1 \end{array}]$

$BA = [\begin{array}{ccc} - 2 & 3 & - 1 \\ - 1 & 2 & - 1 \\ 6 & 9 & - 4 \end{array}] [\begin{array}{ccc} 1 & 3 & - 1 \\ 2 & - 1 & - 1 \\ 3 & 0 & - 1 \end{array}]$

$= [\begin{array}{ccc} 3 & - 9 & 0 \\ 0 & - 5 & 0 \\ 12 & 9 & - 10 \end{array}]$

Hence AB ≠ BA