Algebra Examples

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

Step 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $3$ by its cofactor and add.

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Step 1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 1.3

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$

Step 1.4

Multiply element $a_{31}$ by its cofactor.

$1 | \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$

Step 1.5

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} 4 & 9 \\ 3 & 7 \end{array} |$

Step 1.6

Multiply element $a_{32}$ by its cofactor.

$- 4 | \begin{array}{c} 4 & 9 \\ 3 & 7 \end{array} |$

Step 1.7

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

$| \begin{array}{c} 4 & 3 \\ 3 & 2 \end{array} |$

Step 1.8

Multiply element $a_{33}$ by its cofactor.

$x | \begin{array}{c} 4 & 3 \\ 3 & 2 \end{array} |$

Step 1.9

Add the terms together.

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

Step 2

Evaluate $| \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$ .

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Step 2.1

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

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

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $3$ by $7$ .

$1 (21 - 2 \cdot 9) - 4 | \begin{array}{c} 4 & 9 \\ 3 & 7 \end{array} | + x | \begin{array}{c} 4 & 3 \\ 3 & 2 \end{array} |$

Step 2.2.1.2

Multiply $- 2$ by $9$ .

$1 (21 - 18) - 4 | \begin{array}{c} 4 & 9 \\ 3 & 7 \end{array} | + x | \begin{array}{c} 4 & 3 \\ 3 & 2 \end{array} |$

Step 2.2.2

Subtract $18$ from $21$ .

$1 \cdot 3 - 4 | \begin{array}{c} 4 & 9 \\ 3 & 7 \end{array} | + x | \begin{array}{c} 4 & 3 \\ 3 & 2 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 4 & 9 \\ 3 & 7 \end{array} |$ .

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Step 3.1

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

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

Step 3.2

Simplify the determinant.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $4$ by $7$ .

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

Step 3.2.1.2

Multiply $- 3$ by $9$ .

$1 \cdot 3 - 4 (28 - 27) + x | \begin{array}{c} 4 & 3 \\ 3 & 2 \end{array} |$

Step 3.2.2

Subtract $27$ from $28$ .

$1 \cdot 3 - 4 \cdot 1 + x | \begin{array}{c} 4 & 3 \\ 3 & 2 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 4 & 3 \\ 3 & 2 \end{array} |$ .

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Step 4.1

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

$1 \cdot 3 - 4 \cdot 1 + x (4 \cdot 2 - 3 \cdot 3)$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $4$ by $2$ .

$1 \cdot 3 - 4 \cdot 1 + x (8 - 3 \cdot 3)$

Step 4.2.1.2

Multiply $- 3$ by $3$ .

$1 \cdot 3 - 4 \cdot 1 + x (8 - 9)$

Step 4.2.2

Subtract $9$ from $8$ .

$1 \cdot 3 - 4 \cdot 1 + x \cdot - 1$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $3$ by $1$ .

$3 - 4 \cdot 1 + x \cdot - 1$

Step 5.1.2

Multiply $- 4$ by $1$ .

$3 - 4 + x \cdot - 1$

Step 5.1.3

Move $- 1$ to the left of $x$ .

$3 - 4 - 1 \cdot x$

Step 5.1.4

Rewrite $- 1 x$ as $- x$ .

$3 - 4 - x$

Step 5.2

Subtract $4$ from $3$ .

$- 1 - x$