Find the Determinant [[x,0,-2],[3,x,x^2],[-5,x,1]]

$[\begin{array}{c} x & 0 & - 2 \\ 3 & x & x^{2} \\ - 5 & x & 1 \end{array}]$

Step 1

Multiply every element in the $2 n d$ column by its cofactor and add.

$0 | \begin{array}{c} 3 & x^{2} \\ - 5 & 1 \end{array} | + x | \begin{array}{c} x & - 2 \\ - 5 & 1 \end{array} | - x | \begin{array}{c} x & - 2 \\ 3 & x^{2} \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 3 & x^{2} \\ - 5 & 1 \end{array} |$ .

$0 + x | \begin{array}{c} x & - 2 \\ - 5 & 1 \end{array} | - x | \begin{array}{c} x & - 2 \\ 3 & x^{2} \end{array} |$

Step 3

Evaluate $| \begin{array}{c} x & - 2 \\ - 5 & 1 \end{array} |$ .

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

$0 + x (x \cdot 1 - (- 5 \cdot - 2)) - x | \begin{array}{c} x & - 2 \\ 3 & x^{2} \end{array} |$

Simplify each term.

Multiply $x$ by $1$ .

$0 + x (x - (- 5 \cdot - 2)) - x | \begin{array}{c} x & - 2 \\ 3 & x^{2} \end{array} |$

Multiply $- (- 5 \cdot - 2)$ .

Multiply $- 5$ by $- 2$ .

$0 + x (x - 1 \cdot 10) - x | \begin{array}{c} x & - 2 \\ 3 & x^{2} \end{array} |$

Multiply $- 1$ by $10$ .

$0 + x (x - 10) - x | \begin{array}{c} x & - 2 \\ 3 & x^{2} \end{array} |$

Step 4

Evaluate $| \begin{array}{c} x & - 2 \\ 3 & x^{2} \end{array} |$ .

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

$0 + x (x - 10) - x (x \cdot x^{2} - 3 \cdot - 2)$

Simplify each term.

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$0 + x (x - 10) - x (x^{1} x^{2} - 3 \cdot - 2)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0 + x (x - 10) - x (x^{1 + 2} - 3 \cdot - 2)$

Add $1$ and $2$ .

$0 + x (x - 10) - x (x^{3} - 3 \cdot - 2)$

Multiply $- 3$ by $- 2$ .

$0 + x (x - 10) - x (x^{3} + 6)$

Step 5

Simplify the determinant.

Add $0$ and $x (x - 10)$ .

$x (x - 10) - x (x^{3} + 6)$

Simplify each term.

Apply the distributive property.

$x \cdot x + x \cdot - 10 - x (x^{3} + 6)$

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 10 - x (x^{3} + 6)$

Move $- 10$ to the left of $x$ .

$x^{2} - 10 \cdot x - x (x^{3} + 6)$

Apply the distributive property.

$x^{2} - 10 x - x \cdot x^{3} - x \cdot 6$

Multiply $x$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$x^{2} - 10 x - (x^{3} x) - x \cdot 6$

Multiply $x^{3}$ by $x$ .

Raise $x$ to the power of $1$ .

$x^{2} - 10 x - (x^{3} x^{1}) - x \cdot 6$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} - 10 x - x^{3 + 1} - x \cdot 6$

Add $3$ and $1$ .

$x^{2} - 10 x - x^{4} - x \cdot 6$

Multiply $6$ by $- 1$ .

$x^{2} - 10 x - x^{4} - 6 x$

Subtract $6 x$ from $- 10 x$ .

$x^{2} - x^{4} - 16 x$

Find the Determinant [[x,0,-2],[3,x,x^2],[-5,x,1]]

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