Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 8 & - 7 5 & 7 9 & - 7 \end{matrix} ⎤ ⎥ ⎦ - x = 4 \cdot ⎡ ⎢ ⎣ \begin{matrix} - 1 & - 5 - 6 & 4 - 8 & 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}] - x = 4 \cdot [\begin{array}{c} - 1 & - 5 \\ - 6 & 4 \\ - 8 & 1 \end{array}]$

Step 1

Multiply

4

$4$ by each element of the matrix.

⎡ ⎢ ⎣ \begin{matrix} 8 & - 7 5 & 7 9 & - 7 \end{matrix} ⎤ ⎥ ⎦ - x = ⎡ ⎢ ⎣ \begin{matrix} 4 \cdot - 1 & 4 \cdot - 5 4 \cdot - 6 & 4 \cdot 4 4 \cdot - 8 & 4 \cdot 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}] - x = [\begin{array}{c} 4 \cdot - 1 & 4 \cdot - 5 \\ 4 \cdot - 6 & 4 \cdot 4 \\ 4 \cdot - 8 & 4 \cdot 1 \end{array}]$

Step 2

Simplify each element in the matrix.

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Step 2.1

Multiply

4

$4$ by

- 1

$- 1$ .

⎡ ⎢ ⎣ \begin{matrix} 8 & - 7 5 & 7 9 & - 7 \end{matrix} ⎤ ⎥ ⎦ - x = ⎡ ⎢ ⎣ \begin{matrix} - 4 & 4 \cdot - 5 4 \cdot - 6 & 4 \cdot 4 4 \cdot - 8 & 4 \cdot 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}] - x = [\begin{array}{c} - 4 & 4 \cdot - 5 \\ 4 \cdot - 6 & 4 \cdot 4 \\ 4 \cdot - 8 & 4 \cdot 1 \end{array}]$

Step 2.2

Multiply

4

$4$ by

- 5

$- 5$ .

⎡ ⎢ ⎣ \begin{matrix} 8 & - 7 5 & 7 9 & - 7 \end{matrix} ⎤ ⎥ ⎦ - x = ⎡ ⎢ ⎣ \begin{matrix} - 4 & - 20 4 \cdot - 6 & 4 \cdot 4 4 \cdot - 8 & 4 \cdot 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}] - x = [\begin{array}{c} - 4 & - 20 \\ 4 \cdot - 6 & 4 \cdot 4 \\ 4 \cdot - 8 & 4 \cdot 1 \end{array}]$

Step 2.3

Multiply

4

$4$ by

- 6

$- 6$ .

⎡ ⎢ ⎣ \begin{matrix} 8 & - 7 5 & 7 9 & - 7 \end{matrix} ⎤ ⎥ ⎦ - x = ⎡ ⎢ ⎣ \begin{matrix} - 4 & - 20 - 24 & 4 \cdot 4 4 \cdot - 8 & 4 \cdot 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}] - x = [\begin{array}{c} - 4 & - 20 \\ - 24 & 4 \cdot 4 \\ 4 \cdot - 8 & 4 \cdot 1 \end{array}]$

Step 2.4

Multiply

4

$4$ by

4

$4$ .

⎡ ⎢ ⎣ \begin{matrix} 8 & - 7 5 & 7 9 & - 7 \end{matrix} ⎤ ⎥ ⎦ - x = ⎡ ⎢ ⎣ \begin{matrix} - 4 & - 20 - 24 & 16 4 \cdot - 8 & 4 \cdot 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}] - x = [\begin{array}{c} - 4 & - 20 \\ - 24 & 16 \\ 4 \cdot - 8 & 4 \cdot 1 \end{array}]$

Step 2.5

Multiply

4

$4$ by

- 8

$- 8$ .

⎡ ⎢ ⎣ \begin{matrix} 8 & - 7 5 & 7 9 & - 7 \end{matrix} ⎤ ⎥ ⎦ - x = ⎡ ⎢ ⎣ \begin{matrix} - 4 & - 20 - 24 & 16 - 32 & 4 \cdot 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}] - x = [\begin{array}{c} - 4 & - 20 \\ - 24 & 16 \\ - 32 & 4 \cdot 1 \end{array}]$

Step 2.6

Multiply

4

$4$ by

1

$1$ .

⎡ ⎢ ⎣ \begin{matrix} 8 & - 7 5 & 7 9 & - 7 \end{matrix} ⎤ ⎥ ⎦ - x = ⎡ ⎢ ⎣ \begin{matrix} - 4 & - 20 - 24 & 16 - 32 & 4 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}] - x = [\begin{array}{c} - 4 & - 20 \\ - 24 & 16 \\ - 32 & 4 \end{array}]$

Step 3

Move all terms not containing a variable to the right side.

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Step 3.1

Subtract

$[\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}]$ from both sides of the equation.

$- x = [\begin{array}{c} - 4 & - 20 \\ - 24 & 16 \\ - 32 & 4 \end{array}] - [\begin{array}{c} 8 & - 7 \\ 5 & 7 \\ 9 & - 7 \end{array}]$

Step 4

Simplify the right side.

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Step 4.1

Subtract the corresponding elements.

$- x = [\begin{array}{c} - 4 - 8 & - 20 + 7 \\ - 24 - 5 & 16 - 7 \\ - 32 - 9 & 4 + 7 \end{array}]$

Step 4.2

Simplify each element.

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Step 4.2.1

Subtract

$8$ from

$- 4$ .

$- x = [\begin{array}{c} - 12 & - 20 + 7 \\ - 24 - 5 & 16 - 7 \\ - 32 - 9 & 4 + 7 \end{array}]$

Step 4.2.2

Add

$- 20$ and

$7$ .

$- x = [\begin{array}{c} - 12 & - 13 \\ - 24 - 5 & 16 - 7 \\ - 32 - 9 & 4 + 7 \end{array}]$

Step 4.2.3

Subtract

$5$ from

$- 24$ .

$- x = [\begin{array}{c} - 12 & - 13 \\ - 29 & 16 - 7 \\ - 32 - 9 & 4 + 7 \end{array}]$

Step 4.2.4

Subtract

$7$ from

$16$ .

$- x = [\begin{array}{c} - 12 & - 13 \\ - 29 & 9 \\ - 32 - 9 & 4 + 7 \end{array}]$

Step 4.2.5

Subtract

$9$ from

$- 32$ .

$- x = [\begin{array}{c} - 12 & - 13 \\ - 29 & 9 \\ - 41 & 4 + 7 \end{array}]$

Step 4.2.6

Add

$4$ and

$7$ .

$- x = [\begin{array}{c} - 12 & - 13 \\ - 29 & 9 \\ - 41 & 11 \end{array}]$

Step 5

Multiply both sides by

$- 1$ .

$- - x = - [\begin{array}{c} - 12 & - 13 \\ - 29 & 9 \\ - 41 & 11 \end{array}]$

Step 6

Simplify.

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Step 6.1

Multiply

$- - x$ .

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Step 6.1.1

Multiply

$- 1$ by

$- 1$ .

$1 x = - [\begin{array}{c} - 12 & - 13 \\ - 29 & 9 \\ - 41 & 11 \end{array}]$

Step 6.1.2

Multiply

$x$ by

$1$ .

$x = - [\begin{array}{c} - 12 & - 13 \\ - 29 & 9 \\ - 41 & 11 \end{array}]$

Step 6.2

Multiply

$- 1$ by each element of the matrix.

$x = [\begin{array}{c} - - 12 & - - 13 \\ - - 29 & - 1 \cdot 9 \\ - - 41 & - 1 \cdot 11 \end{array}]$

Step 6.3

Simplify each element in the matrix.

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Step 6.3.1

Multiply

$- 1$ by

$- 12$ .

$x = [\begin{array}{c} 12 & - - 13 \\ - - 29 & - 1 \cdot 9 \\ - - 41 & - 1 \cdot 11 \end{array}]$

Step 6.3.2

Multiply

$- 1$ by

$- 13$ .

$x = [\begin{array}{c} 12 & 13 \\ - - 29 & - 1 \cdot 9 \\ - - 41 & - 1 \cdot 11 \end{array}]$

Step 6.3.3

Multiply

$- 1$ by

$- 29$ .

$x = [\begin{array}{c} 12 & 13 \\ 29 & - 1 \cdot 9 \\ - - 41 & - 1 \cdot 11 \end{array}]$

Step 6.3.4

Multiply

$- 1$ by

$9$ .

$x = [\begin{array}{c} 12 & 13 \\ 29 & - 9 \\ - - 41 & - 1 \cdot 11 \end{array}]$

Step 6.3.5

Multiply

$- 1$ by

$- 41$ .

$x = [\begin{array}{c} 12 & 13 \\ 29 & - 9 \\ 41 & - 1 \cdot 11 \end{array}]$

Step 6.3.6

Multiply

$- 1$ by

$11$ .

$x = [\begin{array}{c} 12 & 13 \\ 29 & - 9 \\ 41 & - 11 \end{array}]$