Matrix Product (Conventional)/Examples/Arbitrary 4

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Example of (Conventional) Matrix Product

$\begin {bmatrix} 1 & 3 & -2 \\ -2 & -6 & 4 \\ 4 & 12 & -8 \end {bmatrix} \begin {bmatrix} 3 & -1 & 2 \\ 3 & 5 & -4 \\ 6 & 7 & -5 \end {bmatrix} = \begin {bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end {bmatrix}$


Proof

\(\ds \begin {bmatrix} 1 & 3 & -2 \\ -2 & -6 & 4 \\ 4 & 12 & -8 \end {bmatrix} \begin {bmatrix} 3 & -1 & 2 \\ 3 & 5 & -4 \\ 6 & 7 & -5 \end {bmatrix}\) \(=\) \(\ds \begin {bmatrix} 1 \times 3 + 3 \times 3 + -2 \times 6 & 1 \times -1 + 3 \times 5 + -2 \times 7 & 1 \times 2 + 3 \times -4 + -2 \times -5 
                        \\ -2 \times 3 + -6 \times 3 +  4 \times 6 & -2 \times -1 + -6 \times 5 +  4 \times 7 & -2 \times 2 + -6 \times -4 +  4 \times -5
                        \\  4 \times 3 + 12 \times 3 + -8 \times 6 &  4 \times -1 + 12 \times 5 + -8 \times 7 &  4 \times 2 + 12 \times -4 + -8 \times -5 \end {bmatrix}\)
\(\ds \) \(=\) \(\ds \begin {bmatrix} 3 + 9 - 12 & -1 + 15 - 14 & 2 - 12 + 10 
                        \\ -6 - 18 + 24 &  2 - 30 + 28 & -4 + 24 - 20
                        \\ 12 + 36 - 48 & -4 + 60 - 56 &  8 - 48 + 40 \end {bmatrix}\)
\(\ds \) \(=\) \(\ds \begin {bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end {bmatrix}\)

$\blacksquare$


Sources

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