Finite Math Examples

$\sqrt{1} [\begin{array}{c} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}]$

Step 1

Any root of $1$ is $1$ .

$1 [\begin{array}{c} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}]$

Step 2

Multiply $1$ by each element of the matrix.

$[\begin{array}{c} 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \end{array}]$

Step 3

Simplify each element in the matrix.

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Step 3.1

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 \cdot 1 & 1 \cdot 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \end{array}]$

Step 3.2

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \cdot 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \end{array}]$

Step 3.3

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \end{array}]$

Step 3.4

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ 1 & 1 \cdot 1 & 1 \cdot 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \end{array}]$

Step 3.5

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ 1 & 1 & 1 \cdot 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \end{array}]$

Step 3.6

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 \cdot 1 & 1 \cdot 1 & 1 \cdot 1 \end{array}]$

Step 3.7

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 \cdot 1 & 1 \cdot 1 \end{array}]$

Step 3.8

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \cdot 1 \end{array}]$

Step 3.9

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}]$

Step 4

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 $1$ by its cofactor and add.

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Step 4.1

Consider the corresponding sign chart.

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

Step 4.2

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

Step 4.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 4.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 4.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 4.6

Multiply element $a_{12}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 4.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 4.8

Multiply element $a_{13}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 4.9

Add the terms together.

$1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$ .

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Step 5.1

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

$1 (1 \cdot 1 - 1 \cdot 1) - 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 5.2

Simplify the determinant.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $1$ by $1$ .

$1 (1 - 1 \cdot 1) - 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 5.2.1.2

Multiply $- 1$ by $1$ .

$1 (1 - 1) - 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 5.2.2

Subtract $1$ from $1$ .

$1 \cdot 0 - 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 6

Evaluate $| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$ .

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Step 6.1

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

$1 \cdot 0 - 1 (1 \cdot 1 - 1 \cdot 1) + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 6.2

Simplify the determinant.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $1$ by $1$ .

$1 \cdot 0 - 1 (1 - 1 \cdot 1) + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 6.2.1.2

Multiply $- 1$ by $1$ .

$1 \cdot 0 - 1 (1 - 1) + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 6.2.2

Subtract $1$ from $1$ .

$1 \cdot 0 - 1 \cdot 0 + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Step 7

Evaluate $| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$ .

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Step 7.1

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

$1 \cdot 0 - 1 \cdot 0 + 1 (1 \cdot 1 - 1 \cdot 1)$

Step 7.2

Simplify the determinant.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $1$ by $1$ .

$1 \cdot 0 - 1 \cdot 0 + 1 (1 - 1 \cdot 1)$

Step 7.2.1.2

Multiply $- 1$ by $1$ .

$1 \cdot 0 - 1 \cdot 0 + 1 (1 - 1)$

Step 7.2.2

Subtract $1$ from $1$ .

$1 \cdot 0 - 1 \cdot 0 + 1 \cdot 0$

Step 8

Simplify the determinant.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $0$ by $1$ .

$0 - 1 \cdot 0 + 1 \cdot 0$

Step 8.1.2

Multiply $- 1$ by $0$ .

$0 + 0 + 1 \cdot 0$

Step 8.1.3

Multiply $0$ by $1$ .

$0 + 0 + 0$

Step 8.2

Add $0$ and $0$ .

$0 + 0$

Step 8.3

Add $0$ and $0$ .

$0$