Compute the products AB and BA whichever exists in the following case: and

Engineering

Mathematics

Algebra of Matrices

Multiplication of Matrices

Question

Compute the products AB and BA whichever exists in the following case:
$A = [\begin{array}{cc} 3 & 2 \\ - 1 & 0 \\ - 1 & 1 \end{array}]$ and $B = [\begin{array}{ccc} 4 & 5 & 6 \\ 0 & 1 & 2 \end{array}]$

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$A = [\begin{array}{cc} 3 & 2 \\ - 1 & 0 \\ - 1 & 1 \end{array}]$ and $B = [\begin{array}{ccc} 4 & 5 & 6 \\ 0 & 1 & 2 \end{array}]$

\Rightarrow AB = [\begin{array}{cc} 3 & 2 \\ - 1 & 0 \\ - 1 & 1 \end{array}] [\begin{array}{ccc} 4 & 5 & 6 \\ 0 & 1 & 2 \end{array}]

= [\begin{array}{ccc} 12 + 0 & 15 + 2 & 18 + 4 \\ - 4 + 0 & - 5 + 0 & - 6 + 0 \\ - 4 + 0 & - 5 + 1 & - 6 + 2 \end{array}]

= [\begin{array}{ccc} 12 & 17 & 22 \\ - 4 & - 5 & - 6 \\ - 4 & - 4 & - 4 \end{array}]

\Rightarrow AB = [\begin{array}{ccc} 12 & 17 & 22 \\ - 4 & - 5 & - 6 \\ - 4 & - 4 & - 4 \end{array}]

\Rightarrow BA = [\begin{array}{ccc} 4 & 5 & 6 \\ 0 & 1 & 2 \end{array}] [\begin{array}{cc} 3 & 2 \\ - 1 & 0 \\ - 1 & 1 \end{array}]

= [\begin{array}{cc} 12 - 5 - 6 & 8 + 0 + 6 \\ 0 - 1 - 2 & 0 + 0 + 2 \end{array}]

= [\begin{array}{cc} 1 & 14 \\ - 3 & 2 \end{array}]

\Rightarrow BA = [\begin{array}{cc} 1 & 14 \\ - 3 & 2 \end{array}]