Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} i + 2 & 5 i + 5 & - 3 i - 2 - 1 & - i - 3 & i + 1 0 & - 1 & 1 \end{matrix} ⎤ ⎥ ⎦ \cdot ⎡ ⎢ ⎣ \begin{matrix} 13 + i - 4 - i - 5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} i + 2 & 5 i + 5 & - 3 i - 2 \\ - 1 & - i - 3 & i + 1 \\ 0 & - 1 & 1 \end{array}] \cdot [\begin{array}{c} 13 + i \\ - 4 - i \\ - 5 \end{array}]$

Step 1

Multiply each row in the first matrix by each column in the second matrix.

⎡ ⎢ ⎣ \begin{matrix} (i + 2) (13 + i) + (5 i + 5) (- 4 - i) + (- 3 i - 2) \cdot - 5 - (13 + i) + (- i - 3) (- 4 - i) + (i + 1) \cdot - 5 0 (13 + i) - (- 4 - i) + 1 \cdot - 5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} (i + 2) (13 + i) + (5 i + 5) (- 4 - i) + (- 3 i - 2) \cdot - 5 \\ - (13 + i) + (- i - 3) (- 4 - i) + (i + 1) \cdot - 5 \\ 0 (13 + i) - (- 4 - i) + 1 \cdot - 5 \end{array}]$

Step 2

Simplify each element of the matrix by multiplying out all the expressions.

⎡ ⎢ ⎣ \begin{matrix} 20 + 5 i - 7 + i - 1 + i \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 20 + 5 i \\ - 7 + i \\ - 1 + i \end{array}]$