Linear Algebra Examples

$[\begin{array}{c} 1 & 4 & 5 & 8 \\ 7 & 9 & 6 & 3 \\ 2 & 5 & 4 & 1 \\ 6 & 5 & 8 & 7 \end{array}]$

Set up the determinant by breaking it into smaller components.

$1 | \begin{array}{c} 9 & 6 & 3 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $1 | \begin{array}{c} 9 & 6 & 3 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} |$ is $60$ .

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Set up the determinant by breaking it into smaller components.

$9 | \begin{array}{c} 4 & 1 \\ 8 & 7 \end{array} | - 5 | \begin{array}{c} 6 & 3 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $9 | \begin{array}{c} 4 & 1 \\ 8 & 7 \end{array} |$ is $180$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$9 ((4) (7) - 8 \cdot 1) - 5 | \begin{array}{c} 6 & 3 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $7$ .

$9 (28 - 8 \cdot 1) - 5 | \begin{array}{c} 6 & 3 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 8$ by $1$ .

$9 (28 - 8) - 5 | \begin{array}{c} 6 & 3 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the expression.

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Subtract $8$ from $28$ .

$9 \cdot 20 - 5 | \begin{array}{c} 6 & 3 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $9$ by $20$ .

$180 - 5 | \begin{array}{c} 6 & 3 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $- 5 | \begin{array}{c} 6 & 3 \\ 8 & 7 \end{array} |$ is $- 90$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$180 - 5 ((6) (7) - 8 \cdot 3) + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $6$ by $7$ .

$180 - 5 (42 - 8 \cdot 3) + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 8$ by $3$ .

$180 - 5 (42 - 24) + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the expression.

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Subtract $24$ from $42$ .

$180 - 5 \cdot 18 + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 5$ by $18$ .

$180 - 90 + 5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $5 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} |$ is $- 30$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$180 - 90 + 5 ((6) (1) - 4 \cdot 3) - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $6$ by $1$ .

$180 - 90 + 5 (6 - 4 \cdot 3) - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 4$ by $3$ .

$180 - 90 + 5 (6 - 12) - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the expression.

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Subtract $12$ from $6$ .

$180 - 90 + 5 \cdot - 6 - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $5$ by $- 6$ .

$180 - 90 - 30 - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Subtract $90$ from $180$ .

$90 - 30 - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Subtract $30$ from $90$ .

$60 - 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} | + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $- 7 | \begin{array}{c} 4 & 5 & 8 \\ 5 & 4 & 1 \\ 5 & 8 & 7 \end{array} |$ is $- 630$ .

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Set up the determinant by breaking it into smaller components.

$60 - 7 (4 | \begin{array}{c} 4 & 1 \\ 8 & 7 \end{array} | - 5 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $4 | \begin{array}{c} 4 & 1 \\ 8 & 7 \end{array} |$ is $80$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$60 - 7 (4 ((4) (7) - 8 \cdot 1) - 5 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $7$ .

$60 - 7 (4 (28 - 8 \cdot 1) - 5 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 8$ by $1$ .

$60 - 7 (4 (28 - 8) - 5 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the expression.

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Subtract $8$ from $28$ .

$60 - 7 (4 \cdot 20 - 5 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $4$ by $20$ .

$60 - 7 (80 - 5 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $- 5 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} |$ is $145$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$60 - 7 (80 - 5 ((5) (7) - 8 \cdot 8) + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $5$ by $7$ .

$60 - 7 (80 - 5 (35 - 8 \cdot 8) + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 8$ by $8$ .

$60 - 7 (80 - 5 (35 - 64) + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the expression.

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Subtract $64$ from $35$ .

$60 - 7 (80 - 5 \cdot - 29 + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 5$ by $- 29$ .

$60 - 7 (80 + 145 + 5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $5 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |$ is $- 135$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$60 - 7 (80 + 145 + 5 ((5) (1) - 4 \cdot 8)) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $5$ by $1$ .

$60 - 7 (80 + 145 + 5 (5 - 4 \cdot 8)) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 4$ by $8$ .

$60 - 7 (80 + 145 + 5 (5 - 32)) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the expression.

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Subtract $32$ from $5$ .

$60 - 7 (80 + 145 + 5 \cdot - 27) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $5$ by $- 27$ .

$60 - 7 (80 + 145 - 135) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Add $80$ and $145$ .

$60 - 7 (225 - 135) + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Subtract $135$ from $225$ .

$60 - 7 \cdot 90 + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 7$ by $90$ .

$60 - 630 + 2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} | - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $2 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 8 & 7 \end{array} |$ is $336$ .

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Set up the determinant by breaking it into smaller components.

$60 - 6302 (4 | \begin{array}{c} 6 & 3 \\ 8 & 7 \end{array} | - 9 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $4 | \begin{array}{c} 6 & 3 \\ 8 & 7 \end{array} |$ is $72$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$60 - 6302 (4 ((6) (7) - 8 \cdot 3) - 9 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $6$ by $7$ .

$60 - 6302 (4 (42 - 8 \cdot 3) - 9 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 8$ by $3$ .

$60 - 6302 (4 (42 - 24) - 9 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the expression.

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Subtract $24$ from $42$ .

$60 - 6302 (4 \cdot 18 - 9 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $4$ by $18$ .

$60 - 6302 (72 - 9 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $- 9 | \begin{array}{c} 5 & 8 \\ 8 & 7 \end{array} |$ is $261$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$60 - 6302 (72 - 9 ((5) (7) - 8 \cdot 8) + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $5$ by $7$ .

$60 - 6302 (72 - 9 (35 - 8 \cdot 8) + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 8$ by $8$ .

$60 - 6302 (72 - 9 (35 - 64) + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the expression.

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Subtract $64$ from $35$ .

$60 - 6302 (72 - 9 \cdot - 29 + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 9$ by $- 29$ .

$60 - 6302 (72 + 261 + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |$ is $- 165$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$60 - 6302 (72 + 261 + 5 ((5) (3) - 6 \cdot 8)) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $5$ by $3$ .

$60 - 6302 (72 + 261 + 5 (15 - 6 \cdot 8)) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $- 6$ by $8$ .

$60 - 6302 (72 + 261 + 5 (15 - 48)) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Simplify the expression.

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Subtract $48$ from $15$ .

$60 - 6302 (72 + 261 + 5 \cdot - 33) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $5$ by $- 33$ .

$60 - 6302 (72 + 261 - 165) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Add $72$ and $261$ .

$60 - 6302 (333 - 165) - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Subtract $165$ from $333$ .

$60 - 6302 \cdot 168 - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

Multiply $2$ by $168$ .

$60 - 630 + 336 - 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$

The determinant of $- 6 | \begin{array}{c} 4 & 5 & 8 \\ 9 & 6 & 3 \\ 5 & 4 & 1 \end{array} |$ is $- 324$ .

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Set up the determinant by breaking it into smaller components.

$60 - 630 + 336 - 6 (4 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} | - 9 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

The determinant of $4 | \begin{array}{c} 6 & 3 \\ 4 & 1 \end{array} |$ is $- 24$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$60 - 630 + 336 - 6 (4 ((6) (1) - 4 \cdot 3) - 9 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

Simplify the determinant.

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Simplify each term.

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Multiply $6$ by $1$ .

$60 - 630 + 336 - 6 (4 (6 - 4 \cdot 3) - 9 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

Multiply $- 4$ by $3$ .

$60 - 630 + 336 - 6 (4 (6 - 12) - 9 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

Simplify the expression.

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Subtract $12$ from $6$ .

$60 - 630 + 336 - 6 (4 \cdot - 6 - 9 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

Multiply $4$ by $- 6$ .

$60 - 630 + 336 - 6 (- 24 - 9 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} | + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

The determinant of $- 9 | \begin{array}{c} 5 & 8 \\ 4 & 1 \end{array} |$ is $243$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$60 - 630 + 336 - 6 (- 24 - 9 ((5) (1) - 4 \cdot 8) + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

Simplify the determinant.

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Simplify each term.

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Multiply $5$ by $1$ .

$60 - 630 + 336 - 6 (- 24 - 9 (5 - 4 \cdot 8) + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

Multiply $- 4$ by $8$ .

$60 - 630 + 336 - 6 (- 24 - 9 (5 - 32) + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

Simplify the expression.

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Subtract $32$ from $5$ .

$60 - 630 + 336 - 6 (- 24 - 9 \cdot - 27 + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

Multiply $- 9$ by $- 27$ .

$60 - 630 + 336 - 6 (- 24 + 243 + 5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |)$

The determinant of $5 | \begin{array}{c} 5 & 8 \\ 6 & 3 \end{array} |$ is $- 165$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$60 - 630 + 336 - 6 (- 24 + 243 + 5 ((5) (3) - 6 \cdot 8))$

Simplify the determinant.

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Simplify each term.

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Multiply $5$ by $3$ .

$60 - 630 + 336 - 6 (- 24 + 243 + 5 (15 - 6 \cdot 8))$

Multiply $- 6$ by $8$ .

$60 - 630 + 336 - 6 (- 24 + 243 + 5 (15 - 48))$

Simplify the expression.

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Subtract $48$ from $15$ .

$60 - 630 + 336 - 6 (- 24 + 243 + 5 \cdot - 33)$

Multiply $5$ by $- 33$ .

$60 - 630 + 336 - 6 (- 24 + 243 - 165)$

Add $- 24$ and $243$ .

$60 - 630 + 336 - 6 (219 - 165)$

Subtract $165$ from $219$ .

$60 - 630 + 336 - 6 \cdot 54$

Multiply $- 6$ by $54$ .

$60 - 630 + 336 - 324$

Subtract $630$ from $60$ .

$- 570 + 336 - 324$

Add $- 570$ and $336$ .

$- 234 - 324$

Subtract $324$ from $- 234$ .

$- 558$

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