Solve {ccc}{1}&{0}&{a}{0}&{a}&{-1}{1}&{2}&{1}

$$\left( \begin{array} { c c c } { 1 } & { 0 } & { a } \\ { 0 } & { a } & { - 1 } \\ { 1 } & { 2 } & { 1 } \end{array} \right)$$

Calculate Determinant

$$\left(-a-1\right)\left(a-2\right)$$

Steps Using Sarrus' Rule

Find the determinant of the matrix using the method of diagonals.

$$det(\left(\begin{matrix}1&0&a\\0&a&-1\\1&2&1\end{matrix}\right))$$

Extend the original matrix by repeating the first two columns as the fourth and fifth columns.

$$\left(\begin{matrix}1&0&a&1&0\\0&a&-1&0&a\\1&2&1&1&2\end{matrix}\right)$$

Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.

$$a=a$$

Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.

$$aa+2\left(-1\right)=a^{2}-2$$

Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.

$$a-\left(a^{2}-2\right)$$

Subtract a^{2}-2 from a.

$$-\left(a-2\right)\left(a+1\right)$$

Steps Using Expansion by Minors

Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).

$$det(\left(\begin{matrix}1&0&a\\0&a&-1\\1&2&1\end{matrix}\right))$$

To expand by minors, multiply each element of the first row by its minor, which is the determinant of the 2\times 2 matrix created by deleting the row and column containing that element, then multiply by the element's position sign.

$$det(\left(\begin{matrix}a&-1\\2&1\end{matrix}\right))+adet(\left(\begin{matrix}0&a\\1&2\end{matrix}\right))$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.

$$a-2\left(-1\right)+a\left(-a\right)$$

Simplify.

$$a+2+a\left(-a\right)$$

Add the terms to obtain the final result.

$$-\left(a-2\right)\left(a+1\right)$$

Evaluate

$$\left(\begin{matrix}1&0&a\\0&a&-1\\1&2&1\end{matrix}\right)$$