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.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

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 \).

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...