Matrix Product (Conventional)/Examples/Arbitrary 1

Example of (Conventional) Matrix Product

$\begin {bmatrix} 2 & 1 & 0 \\ 3 & 0 & 7 \end {bmatrix} \begin {bmatrix} 2 & 3 & 5 & 8 \\ 4 & 8 & 6 & 1 \\ -1 & 7 & 0 & 7 \end {bmatrix} = \begin {bmatrix} 8 & 14 & 16 & 17 \\ -1 & 58 & 15 & 73 \end {bmatrix}$

Proof

\(\ds \begin {bmatrix} 2 & 1 & 0 \\ 3 & 0 & 7 \end {bmatrix} \begin {bmatrix} 2 & 3 & 5 & 8 \\ 4 & 8 & 6 & 1 \\ -1 & 7 & 0 & 7 \end {bmatrix}\)

\(=\)

\(\ds \begin {bmatrix} 2 \times 2 + 1 \times 4 + 0 \times -1 & 2 \times 3 + 1 \times 8 + 0 \times 7 & 2 \times 5 + 1 \times 6 + 0 \times 0 & 2 \times 8 + 1 \times 1 + 0 \times 7 \\ 3 \times 2 + 0 \times 4 + 7 \times -1 & 3 \times 3 + 0 \times 8 + 7 \times 7 & 3 \times 5 + 0 \times 6 + 7 \times 0 & 3 \times 8 + 0 \times 1 + 7 \times 7 \end {bmatrix}\)

\(\ds \)

\(=\)

\(\ds \begin {bmatrix} 4 + 4 + 0 & 6 + 8 + 0 & 10 + 6 + 0 & 16 + 1 + 0 \\ 6 + 0 - 7 & 9 + 0 + 49 & 15 + 0 + 0 & 24 + 0 + 49 \end {bmatrix}\)

\(\ds \)

\(=\)

\(\ds \begin {bmatrix} 8 & 14 & 16 & 17 \\ -1 & 58 & 15 & 73 \end {bmatrix}\)

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices