If A = begin{bmatrix} 1 & -2 & 3 -4 & 2 & 5 | vedclass

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Since $A$ is a $2 	imes 3$ matrix and $B$ is a $3 	imes 2$ matrix,both $AB$ and $BA$ are defined. The order of $AB$ is $2 	imes 2$ and the order of

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Since $A$ is a $2 	imes 3$ matrix and $B$ is a $3 	imes 2$ matrix,both $AB$ and $BA$ are defined. The order of $AB$ is $2 	imes 2$ and the order of $BA$ is $3 	imes 3$.$AB = \begin{bmatrix} 1 & -2 & 3 \ -4 & 2 & 5 \end{bmatrix} \begin{bmatrix} 2 & 3 \ 4 & 5 \ 2 & 1 \end{bmatrix} = \begin{bmatrix} (1)(2) + (-2)(4) + (3)(2) & (1)(3) + (-2)(5) + (3)(1) \ (-4)(2) + (2)(4) + (5)(2) & (-4)(3) + (2)(5) + (5)(1) \end{bmatrix} = \begin{bmatrix} 2 - 8 + 6 & 3 - 10 + 3 \ -8 + 8 + 10 & -12 + 10 + 5 \end{bmatrix} = \begin{bmatrix} 0 & -4 \ 10 & 3 \end{bmatrix}$.$BA = \begin{bmatrix} 2 & 3 \ 4 & 5 \ 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & -2 & 3 \ -4 & 2 & 5 \end{bmatrix} = \begin{bmatrix} (2)(1) + (3)(-4) & (2)(-2) + (3)(2) & (2)(3) + (3)(5) \ (4)(1) + (5)(-4) & (4)(-2) + (5)(2) & (4)(3) + (5)(5) \ (2)(1) + (1)(-4) & (2)(-2) + (1)(2) & (2)(3) + (1)(5) \end{bmatrix} = \begin{bmatrix} 2 - 12 & -4 + 6 & 6 + 15 \ 4 - 20 & -8 + 10 & 12 + 25 \ 2 - 4 & -4 + 2 & 6 + 5 \end{bmatrix} = \begin{bmatrix} -10 & 2 & 21 \ -16 & 2 & 37 \ -2 & -2 & 11 \end{bmatrix}$.Since $AB$ is a $2 	imes 2$ matrix and $BA$ is a $3 	imes 3$ matrix,clearly $AB 
eq BA$.

If $A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix}$,then find $AB$ and $BA$. Show that $AB \neq BA$.

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