Linear Algebra Examples

$[\begin{array}{c} - 3 i & 2 j & 4 k \\ 5 i & 1 j & 6 k \\ 2 i & 3 j & 1 k \end{array}]$

Step 1

Multiply $j$ by $1$ .

$[\begin{array}{c} - 3 i & 2 j & 4 k \\ 5 i & j & 6 k \\ 2 i & 3 j & 1 k \end{array}]$

Step 2

Multiply $k$ by $1$ .

$[\begin{array}{c} - 3 i & 2 j & 4 k \\ 5 i & j & 6 k \\ 2 i & 3 j & k \end{array}]$

Step 3

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 3.1

Consider the corresponding sign chart.

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

Step 3.2

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

Step 3.3

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

$| \begin{array}{c} j & 6 k \\ 3 j & k \end{array} |$

Step 3.4

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

$- 3 i | \begin{array}{c} j & 6 k \\ 3 j & k \end{array} |$

Step 3.5

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

$| \begin{array}{c} 5 i & 6 k \\ 2 i & k \end{array} |$

Step 3.6

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

$- 2 j | \begin{array}{c} 5 i & 6 k \\ 2 i & k \end{array} |$

Step 3.7

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

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

Step 3.8

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

$4 k | \begin{array}{c} 5 i & j \\ 2 i & 3 j \end{array} |$

Step 3.9

Add the terms together.

$- 3 i | \begin{array}{c} j & 6 k \\ 3 j & k \end{array} | - 2 j | \begin{array}{c} 5 i & 6 k \\ 2 i & k \end{array} | + 4 k | \begin{array}{c} 5 i & j \\ 2 i & 3 j \end{array} |$

Step 4

Evaluate $| \begin{array}{c} j & 6 k \\ 3 j & k \end{array} |$ .

Tap for more steps...

Step 4.1

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

$- 3 i (j k - 3 j (6 k)) - 2 j | \begin{array}{c} 5 i & 6 k \\ 2 i & k \end{array} | + 4 k | \begin{array}{c} 5 i & j \\ 2 i & 3 j \end{array} |$

Step 4.2

Simplify the determinant.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Rewrite using the commutative property of multiplication.

$- 3 i (j k - 3 \cdot 6 j k) - 2 j | \begin{array}{c} 5 i & 6 k \\ 2 i & k \end{array} | + 4 k | \begin{array}{c} 5 i & j \\ 2 i & 3 j \end{array} |$

Step 4.2.1.2

Multiply $- 3$ by $6$ .

$- 3 i (j k - 18 j k) - 2 j | \begin{array}{c} 5 i & 6 k \\ 2 i & k \end{array} | + 4 k | \begin{array}{c} 5 i & j \\ 2 i & 3 j \end{array} |$

Step 4.2.2

Subtract $18 j k$ from $j k$ .

$- 3 i (- 17 j k) - 2 j | \begin{array}{c} 5 i & 6 k \\ 2 i & k \end{array} | + 4 k | \begin{array}{c} 5 i & j \\ 2 i & 3 j \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 5 i & 6 k \\ 2 i & k \end{array} |$ .

Tap for more steps...

Step 5.1

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

$- 3 i (- 17 j k) - 2 j (5 i k - 2 i (6 k)) + 4 k | \begin{array}{c} 5 i & j \\ 2 i & 3 j \end{array} |$

Step 5.2

Simplify the determinant.

Tap for more steps...

Step 5.2.1

Multiply $6$ by $- 2$ .

$- 3 i (- 17 j k) - 2 j (5 i k - 12 i k) + 4 k | \begin{array}{c} 5 i & j \\ 2 i & 3 j \end{array} |$

Step 5.2.2

Subtract $12 i k$ from $5 i k$ .

$- 3 i (- 17 j k) - 2 j (- 7 i k) + 4 k | \begin{array}{c} 5 i & j \\ 2 i & 3 j \end{array} |$

Step 6

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

Tap for more steps...

Step 6.1

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

$- 3 i (- 17 j k) - 2 j (- 7 i k) + 4 k (5 i (3 j) - 2 i j)$

Step 6.2

Simplify the determinant.

Tap for more steps...

Step 6.2.1

Multiply $3$ by $5$ .

$- 3 i (- 17 j k) - 2 j (- 7 i k) + 4 k (15 i j - 2 i j)$

Step 6.2.2

Subtract $2 i j$ from $15 i j$ .

$- 3 i (- 17 j k) - 2 j (- 7 i k) + 4 k (13 i j)$

Step 7

Simplify the determinant.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Multiply $- 17$ by $- 3$ .

$51 i (j k) - 2 j (- 7 i k) + 4 k (13 i j)$

Step 7.1.2

Rewrite using the commutative property of multiplication.

$51 i j k - 2 \cdot - 7 i j k + 4 k (13 i j)$

Step 7.1.3

Multiply $- 2$ by $- 7$ .

$51 i j k + 14 i j k + 4 k (13 i j)$

Step 7.1.4

Rewrite using the commutative property of multiplication.

$51 i j k + 14 i j k + 4 \cdot 13 i k j$

Step 7.1.5

Multiply $4$ by $13$ .

$51 i j k + 14 i j k + 52 i k j$

Step 7.2

Add $51 i j k$ and $14 i j k$ .

$65 i j k + 52 i k j$

Step 7.3

Move $k$ .

$65 j k i + 52 j k i$

Step 7.4

Add $65 j k i$ and $52 j k i$ .

$117 j k i$