Linear Algebra Examples

$i [\begin{array}{c} - 8 & 9 \\ 6 & - 9 \end{array}] - j [\begin{array}{c} 8 & 9 \\ 65 & - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $i$ by each element of the matrix.

$[\begin{array}{c} i \cdot - 8 & i \cdot 9 \\ i \cdot 6 & i \cdot - 9 \end{array}] - j [\begin{array}{c} 8 & 9 \\ 65 & - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Move $- 8$ to the left of $i$ .

$[\begin{array}{c} - 8 i & i \cdot 9 \\ i \cdot 6 & i \cdot - 9 \end{array}] - j [\begin{array}{c} 8 & 9 \\ 65 & - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.2.2

Move $9$ to the left of $i$ .

$[\begin{array}{c} - 8 i & 9 i \\ i \cdot 6 & i \cdot - 9 \end{array}] - j [\begin{array}{c} 8 & 9 \\ 65 & - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.2.3

Move $6$ to the left of $i$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & i \cdot - 9 \end{array}] - j [\begin{array}{c} 8 & 9 \\ 65 & - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.2.4

Move $- 9$ to the left of $i$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] - j [\begin{array}{c} 8 & 9 \\ 65 & - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.3

Multiply $- j$ by each element of the matrix.

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - j \cdot 8 & - j \cdot 9 \\ - j \cdot 65 & - j \cdot - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.4

Simplify each element in the matrix.

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Step 1.4.1

Multiply $8$ by $- 1$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - 8 j & - j \cdot 9 \\ - j \cdot 65 & - j \cdot - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.4.2

Multiply $9$ by $- 1$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - 8 j & - 9 j \\ - j \cdot 65 & - j \cdot - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.4.3

Multiply $65$ by $- 1$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - 8 j & - 9 j \\ - 65 j & - j \cdot - 14 \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.4.4

Multiply $- 14$ by $- 1$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - 8 j & - 9 j \\ - 65 j & 14 j \end{array}] + k [\begin{array}{c} - 9 & 5 \\ - 3 & - 8 \end{array}]$

Step 1.5

Multiply $k$ by each element of the matrix.

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - 8 j & - 9 j \\ - 65 j & 14 j \end{array}] + [\begin{array}{c} k \cdot - 9 & k \cdot 5 \\ k \cdot - 3 & k \cdot - 8 \end{array}]$

Step 1.6

Simplify each element in the matrix.

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Step 1.6.1

Move $- 9$ to the left of $k$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - 8 j & - 9 j \\ - 65 j & 14 j \end{array}] + [\begin{array}{c} - 9 k & k \cdot 5 \\ k \cdot - 3 & k \cdot - 8 \end{array}]$

Step 1.6.2

Move $5$ to the left of $k$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - 8 j & - 9 j \\ - 65 j & 14 j \end{array}] + [\begin{array}{c} - 9 k & 5 k \\ k \cdot - 3 & k \cdot - 8 \end{array}]$

Step 1.6.3

Move $- 3$ to the left of $k$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - 8 j & - 9 j \\ - 65 j & 14 j \end{array}] + [\begin{array}{c} - 9 k & 5 k \\ - 3 k & k \cdot - 8 \end{array}]$

Step 1.6.4

Move $- 8$ to the left of $k$ .

$[\begin{array}{c} - 8 i & 9 i \\ 6 i & - 9 i \end{array}] + [\begin{array}{c} - 8 j & - 9 j \\ - 65 j & 14 j \end{array}] + [\begin{array}{c} - 9 k & 5 k \\ - 3 k & - 8 k \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} - 8 i - 8 j & 9 i - 9 j \\ 6 i - 65 j & - 9 i + 14 j \end{array}] + [\begin{array}{c} - 9 k & 5 k \\ - 3 k & - 8 k \end{array}]$

Step 3

Add the corresponding elements.

$[\begin{array}{c} - 8 i - 8 j - 9 k & 9 i - 9 j + 5 k \\ 6 i - 65 j - 3 k & - 9 i + 14 j - 8 k \end{array}]$