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# True or False: If $A =\begin{bmatrix} 2 & 3 & 1 \\ 1 &4 & 2 \end{bmatrix}\; and\; B=\begin{bmatrix}2 & 3 \\ 4& 5 \\ 2 & 1 \end{bmatrix},$ then AB and BA are defined and equal.

In this question, AB and BA need not be evaluated. While AB is a 2 X 2 matrix, BA is a 3 X 3 matrix by definition of matrix multiplication. By definition of equality of two matrices, since they are not of the same type, AB is not equal to BA. So the statement is false. This could be taken as an alternate solution.

Toolbox:
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Step1:
Given
$A =\begin{bmatrix} 2 & 3 & 1 \\ 1 &4 & 2 \end{bmatrix}$
$B=\begin{bmatrix}2 & 3 \\ 4& 5 \\ 2 & 1 \end{bmatrix}$
$AB=\begin{bmatrix} 2 & 3 & 1 \\ 1 &4 & 2 \end{bmatrix} \begin{bmatrix}2 & 3 \\ 4& 5 \\ 2 & 1 \end{bmatrix}$,
$AB=\begin{bmatrix}2(2)+3(4)+1(2) & 2(3)+3(5)+1(1) \\ 1(2)+4(4)+2(2)& 1(3)+4(5)+2(1) \end{bmatrix}$
$AB=\begin{bmatrix}4+12+2 & 6+15+1 \\ 2+16+4& 3+20+2 \end{bmatrix}$
$AB=\begin{bmatrix}18 & 22 \\ 22& 25 \end{bmatrix}$
Step2:
$BA=\begin{bmatrix}2 & 3 \\ 4& 5 \\ 2 & 1 \end{bmatrix}\begin{bmatrix} 2 & 3 & 1 \\ 1 &4 & 2 \end{bmatrix}$
$BA =\begin{bmatrix} 2(2)+3(1) & 2(3)+3(4) & 2(1)+3(2) \\ 4(2)+5(1) &4(3)+5(4) & 4(1)+5(2) \\2(2)+1(1) &2(3)+1(4)&2(1)+1(2) \end{bmatrix}$
$BA =\begin{bmatrix} 4+3 & 6+12 & 2+6 \\ 8+5 &12+20 & 4+10 \\4+1 &6+4&2+2 \end{bmatrix}$
$BA =\begin{bmatrix} 7 & 18 & 8 \\ 13 &32 & 14 \\5 &10&4 \end{bmatrix}$
Hence AB $\neq$ BA
Thus its False