Find the Determinant [[1,a,0],[1,0,b],[0,c,1]]

$[\begin{array}{c} 1 & a & 0 \\ 1 & 0 & b \\ 0 & c & 1 \end{array}]$

Step 1

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

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

Step 2

Multiply $0$ by $| \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} |$ .

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

Step 3

Evaluate $| \begin{array}{c} 1 & b \\ 0 & 1 \end{array} |$ .

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

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

Simplify the determinant.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $0$ by $b$ .

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

Add $1$ and $0$ .

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

Step 4

Evaluate $| \begin{array}{c} 1 & 0 \\ 1 & b \end{array} |$ .

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

$- a \cdot 1 + 0 - c (1 b - 1 \cdot 0)$

Simplify the determinant.

Multiply $b$ by $1$ .

$- a \cdot 1 + 0 - c (b - 1 \cdot 0)$

Subtract $0$ from $b$ .

$- a \cdot 1 + 0 - c b$

Step 5

Simplify the determinant.

Add $- a \cdot 1$ and $0$ .

$- a \cdot 1 - c b$

Multiply $- 1$ by $1$ .

$- a - c b$

Find the Determinant [[1,a,0],[1,0,b],[0,c,1]]

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