Question

If $A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 2& 3& 4\end{array}\right]B=\left[\begin{array}{cc}-1& -2\\ -1& 0\\ 2& -1\end{array}\right]$ then third element of second column of AB = _______

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Answer 1

We know that two matrices can be multiplied only when number of columns of A = no. of rows of B.

(for AB to be defined).In the above question the 2 matrices satisfy this condition.

Also elements of AB are obtained by adding the products of corresponding elements of $i^{th}$ row of A and $j^{th}$ column of B. This will give $(I,j{)}^{th}$ element of AB.

So $\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 2& 3& 4\end{array}\right]\left[\begin{array}{cc}-1& -2\\ -1& 0\\ 2& -1\end{array}\right]$

$=\left[\begin{array}{cc}(0\times 1)+(1\times -1)+2\times 2& (0\times -2)+(1\times 0)+(2x-1)\\ (1\times 1)+(2\times -1)+3\times 2& (1\times -2)+(2x\times 0)+(3x-1)\\ (2\times 1)+(3x-1)+4\times 2& (2x-2)+(3\times 0)+(4x-1)\end{array}\right]$

$=\left[\begin{array}{cc}3& -2\\ 5& -5\\ 7& -8\end{array}\right]$ So the third element of second column of this matrix will be -8 which is the correct answer.