Find the Determinant [[4,3,9],[3,2,7],[1,4,x]]

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

Set up the determinant by breaking it into smaller components.

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

Find the determinant of $4 | \begin{array}{c} 2 & 7 \\ 4 & x \end{array} |$ .

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

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

Simplify the determinant.

Multiply $- 4$ by $7$ .

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

Simplify by multiplying through.

Apply the distributive property.

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

Multiply.

Multiply $2$ by $4$ .

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

Multiply $4$ by $- 28$ .

$8 x - 112 - 3 | \begin{array}{c} 3 & 9 \\ 4 & x \end{array} | + 1 | \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$

Find the determinant of $- 3 | \begin{array}{c} 3 & 9 \\ 4 & x \end{array} |$ .

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

$8 x - 112 - 3 ((3) (x) - 4 \cdot 9) + 1 | \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$

Simplify the determinant.

Multiply $- 4$ by $9$ .

$8 x - 112 - 3 (3 x - 36) + 1 | \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$

Simplify by multiplying through.

Apply the distributive property.

$8 x - 112 - 3 (3 x) - 3 \cdot - 36 + 1 | \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$

Multiply.

Multiply $3$ by $- 3$ .

$8 x - 112 - 9 x - 3 \cdot - 36 + 1 | \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$

Multiply $- 3$ by $- 36$ .

$8 x - 112 - 9 x + 108 + 1 | \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$

Find the determinant of $1 | \begin{array}{c} 3 & 9 \\ 2 & 7 \end{array} |$ .

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

$8 x - 112 - 9 x + 108 + (3) (7) - 2 \cdot 9$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $7$ .

$8 x - 112 - 9 x + 108 + 21 - 2 \cdot 9$

Multiply $- 2$ by $9$ .

$8 x - 112 - 9 x + 108 + 21 - 18$

Subtract $18$ from $21$ .

$8 x - 112 - 9 x + 108 + 3$

Subtract $9 x$ from $8 x$ .

$- x - 112 + 108 + 3$

Simplify by adding numbers.

Add $- 112$ and $108$ .

$- x - 4 + 3$

Add $- 4$ and $3$ .

$- x - 1$

Find the Determinant [[4,3,9],[3,2,7],[1,4,x]]

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