Matrix Multiplication

Algebra Level 3

Let A , B A, B and C C be matrices such that A = ( 1 2 3 1 2 3 ) A = \begin{pmatrix} 1 & 2 \\ 3 & 1 \\ 2 & 3 \end{pmatrix} , B = ( 1 2 2 2 4 3 ) B = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 4 & 3 \end{pmatrix} and C = A B C=AB .

What is the sum of all the elements (entries) of matrix C C ?


The answer is 84.

Arron Kau by Arron Kau

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1 solution

Arron Kau Staff
May 13, 2014

Solution 1: We have C = A B = ( 1 ( 1 ) + 2 ( 2 ) 2 ( 1 ) + 4 ( 2 ) 2 ( 1 ) + 3 ( 2 ) 1 ( 3 ) + 2 ( 1 ) 2 ( 3 ) + 4 ( 1 ) 2 ( 3 ) + 3 ( 1 ) 1 ( 2 ) + 2 ( 3 ) 2 ( 2 ) + 4 ( 3 ) 2 ( 2 ) + 3 ( 3 ) ) = ( 5 10 8 5 10 9 8 16 13 ) \begin{aligned} C=AB &= \begin{pmatrix} 1(1) + 2(2) & 2(1) + 4(2) & 2(1) + 3(2) \\ 1(3) + 2(1) &2(3) + 4(1) & 2(3) + 3(1) \\ 1(2) + 2(3) & 2(2) + 4(3) & 2(2) + 3(3) \end{pmatrix} \\ &= \begin{pmatrix} 5 & 10 & 8 \\ 5 & 10 & 9 \\ 8 & 16 & 13 \end{pmatrix} \\ \end{aligned}

Hence the sum of the elements is 5 + 10 + 8 + 5 + 10 + 9 + 8 + 16 + 13 = 84 5 + 10 + 8 + 5 + 10 + 9 + 8 + 16 + 13 = 84 .

Solution 2: The sum of all the elements of matrix C C is equal to \( \begin{pmatrix} 1 & 1 & 1 \\ \end{pmatrix} C \begin{pmatrix} 1\\ 1\\ 1\\

\end{pmatrix}

\begin{pmatrix} 1 & 1 & 1 \\ \end{pmatrix} AB \begin{pmatrix} 1\\ 1\\ 1\\

\end{pmatrix}

\begin{pmatrix} 1+3+2 & 2+1+3\\ \end{pmatrix} \cdot \begin{pmatrix} 1+2+2\\ 2+4+3\\ \end{pmatrix} = 84 \).

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