Let and be matrices such that , and .
What is the sum of all the elements (entries) of matrix ?
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Solution 1: We have C = A B = ⎝ ⎛ 1 ( 1 ) + 2 ( 2 ) 1 ( 3 ) + 2 ( 1 ) 1 ( 2 ) + 2 ( 3 ) 2 ( 1 ) + 4 ( 2 ) 2 ( 3 ) + 4 ( 1 ) 2 ( 2 ) + 4 ( 3 ) 2 ( 1 ) + 3 ( 2 ) 2 ( 3 ) + 3 ( 1 ) 2 ( 2 ) + 3 ( 3 ) ⎠ ⎞ = ⎝ ⎛ 5 5 8 1 0 1 0 1 6 8 9 1 3 ⎠ ⎞
Hence the sum of the elements is 5 + 1 0 + 8 + 5 + 1 0 + 9 + 8 + 1 6 + 1 3 = 8 4 .
Solution 2: The sum of all the elements of matrix C is equal to \( \begin{pmatrix} 1 & 1 & 1 \\ \end{pmatrix} C \begin{pmatrix} 1\\ 1\\ 1\\
\end{pmatrix}
\begin{pmatrix} 1 & 1 & 1 \\ \end{pmatrix} AB \begin{pmatrix} 1\\ 1\\ 1\\
\end{pmatrix}
\begin{pmatrix} 1+3+2 & 2+1+3\\ \end{pmatrix} \cdot \begin{pmatrix} 1+2+2\\ 2+4+3\\ \end{pmatrix} = 84 \).