Linear Algebra Examples

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

Step 1

Find the determinant.

Tap for more steps...

Step 1.1

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 $1$ by its cofactor and add.

Tap for more steps...

Step 1.1.1

Consider the corresponding sign chart.

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

Step 1.1.2

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

Step 1.1.3

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

$| \begin{array}{c} 5 & 7 \\ 7 & 9 \end{array} |$

Step 1.1.4

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

$1 | \begin{array}{c} 5 & 7 \\ 7 & 9 \end{array} |$

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.1.7

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

$| \begin{array}{c} 2 & 5 \\ 3 & 7 \end{array} |$

Step 1.1.8

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

$3 | \begin{array}{c} 2 & 5 \\ 3 & 7 \end{array} |$

Step 1.1.9

Add the terms together.

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

Step 1.2

Evaluate $| \begin{array}{c} 5 & 7 \\ 7 & 9 \end{array} |$ .

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Simplify the determinant.

Tap for more steps...

Step 1.2.2.1

Simplify each term.

Tap for more steps...

Step 1.2.2.1.1

Multiply $5$ by $9$ .

$1 (45 - 7 \cdot 7) - 2 | \begin{array}{c} 2 & 7 \\ 3 & 9 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 3 & 7 \end{array} |$

Step 1.2.2.1.2

Multiply $- 7$ by $7$ .

$1 (45 - 49) - 2 | \begin{array}{c} 2 & 7 \\ 3 & 9 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 3 & 7 \end{array} |$

Step 1.2.2.2

Subtract $49$ from $45$ .

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

Simplify the determinant.

Tap for more steps...

Step 1.3.2.1

Simplify each term.

Tap for more steps...

Step 1.3.2.1.1

Multiply $2$ by $9$ .

$1 \cdot - 4 - 2 (18 - 3 \cdot 7) + 3 | \begin{array}{c} 2 & 5 \\ 3 & 7 \end{array} |$

Step 1.3.2.1.2

Multiply $- 3$ by $7$ .

$1 \cdot - 4 - 2 (18 - 21) + 3 | \begin{array}{c} 2 & 5 \\ 3 & 7 \end{array} |$

Step 1.3.2.2

Subtract $21$ from $18$ .

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

Step 1.4

Evaluate $| \begin{array}{c} 2 & 5 \\ 3 & 7 \end{array} |$ .

Tap for more steps...

Step 1.4.1

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

$1 \cdot - 4 - 2 \cdot - 3 + 3 (2 \cdot 7 - 3 \cdot 5)$

Step 1.4.2

Simplify the determinant.

Tap for more steps...

Step 1.4.2.1

Simplify each term.

Tap for more steps...

Step 1.4.2.1.1

Multiply $2$ by $7$ .

$1 \cdot - 4 - 2 \cdot - 3 + 3 (14 - 3 \cdot 5)$

Step 1.4.2.1.2

Multiply $- 3$ by $5$ .

$1 \cdot - 4 - 2 \cdot - 3 + 3 (14 - 15)$

Step 1.4.2.2

Subtract $15$ from $14$ .

$1 \cdot - 4 - 2 \cdot - 3 + 3 \cdot - 1$

Step 1.5

Simplify the determinant.

Tap for more steps...

Step 1.5.1

Simplify each term.

Tap for more steps...

Step 1.5.1.1

Multiply $- 4$ by $1$ .

$- 4 - 2 \cdot - 3 + 3 \cdot - 1$

Step 1.5.1.2

Multiply $- 2$ by $- 3$ .

$- 4 + 6 + 3 \cdot - 1$

Step 1.5.1.3

Multiply $3$ by $- 1$ .

$- 4 + 6 - 3$

Step 1.5.2

Add $- 4$ and $6$ .

$2 - 3$

Step 1.5.3

Subtract $3$ from $2$ .

$- 1$

Step 2

Since the determinant is non-zero, the inverse exists.

Step 3

Set up a $3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 2 & 5 & 7 & 0 & 1 & 0 \\ 3 & 7 & 9 & 0 & 0 & 1 \end{array}]$

Step 4

Find the reduced row echelon form.

Tap for more steps...

Step 4.1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

Tap for more steps...

Step 4.1.1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 2 - 2 \cdot 1 & 5 - 2 \cdot 2 & 7 - 2 \cdot 3 & 0 - 2 \cdot 1 & 1 - 2 \cdot 0 & 0 - 2 \cdot 0 \\ 3 & 7 & 9 & 0 & 0 & 1 \end{array}]$

Step 4.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & 1 & - 2 & 1 & 0 \\ 3 & 7 & 9 & 0 & 0 & 1 \end{array}]$

Step 4.2

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

Tap for more steps...

Step 4.2.1

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & 1 & - 2 & 1 & 0 \\ 3 - 3 \cdot 1 & 7 - 3 \cdot 2 & 9 - 3 \cdot 3 & 0 - 3 \cdot 1 & 0 - 3 \cdot 0 & 1 - 3 \cdot 0 \end{array}]$

Step 4.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & 1 & - 2 & 1 & 0 \\ 0 & 1 & 0 & - 3 & 0 & 1 \end{array}]$

Step 4.3

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

Tap for more steps...

Step 4.3.1

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & 1 & - 2 & 1 & 0 \\ 0 - 0 & 1 - 1 & 0 - 1 & - 3 + 2 & 0 - 1 & 1 - 0 \end{array}]$

Step 4.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & 1 & - 2 & 1 & 0 \\ 0 & 0 & - 1 & - 1 & - 1 & 1 \end{array}]$

Step 4.4

Multiply each element of $R_{3}$ by $- 1$ to make the entry at $3, 3$ a $1$ .

Tap for more steps...

Step 4.4.1

Multiply each element of $R_{3}$ by $- 1$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & 1 & - 2 & 1 & 0 \\ - 0 & - 0 & - - 1 & - - 1 & - - 1 & - 1 \cdot 1 \end{array}]$

Step 4.4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & 1 & - 2 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 & - 1 \end{array}]$

Step 4.5

Perform the row operation $R_{2} = R_{2} - R_{3}$ to make the entry at $2, 3$ a $0$ .

Tap for more steps...

Step 4.5.1

Perform the row operation $R_{2} = R_{2} - R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 - 0 & 1 - 0 & 1 - 1 & - 2 - 1 & 1 - 1 & 0 + 1 \\ 0 & 0 & 1 & 1 & 1 & - 1 \end{array}]$

Step 4.5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & 0 & - 3 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 & - 1 \end{array}]$

Step 4.6

Perform the row operation $R_{1} = R_{1} - 3 R_{3}$ to make the entry at $1, 3$ a $0$ .

Tap for more steps...

Step 4.6.1

Perform the row operation $R_{1} = R_{1} - 3 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 3 \cdot 0 & 2 - 3 \cdot 0 & 3 - 3 \cdot 1 & 1 - 3 \cdot 1 & 0 - 3 \cdot 1 & 0 - 3 \cdot - 1 \\ 0 & 1 & 0 & - 3 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 & - 1 \end{array}]$

Step 4.6.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 & - 2 & - 3 & 3 \\ 0 & 1 & 0 & - 3 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 & - 1 \end{array}]$

Step 4.7

Perform the row operation $R_{1} = R_{1} - 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

Tap for more steps...

Step 4.7.1

Perform the row operation $R_{1} = R_{1} - 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & - 2 - 2 \cdot - 3 & - 3 - 2 \cdot 0 & 3 - 2 \cdot 1 \\ 0 & 1 & 0 & - 3 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 & - 1 \end{array}]$

Step 4.7.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 4 & - 3 & 1 \\ 0 & 1 & 0 & - 3 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 & - 1 \end{array}]$

Step 5

The right half of the reduced row echelon form is the inverse.

$[\begin{array}{c} 4 & - 3 & 1 \\ - 3 & 0 & 1 \\ 1 & 1 & - 1 \end{array}]$