Solve {ccc}{1}&{1}&{-2}{3}&{1}&{a}{a}&{5}&{-1}

$$\left( \begin{array} { c c c } { 1 } & { 1 } & { - 2 } \\ { 3 } & { 1 } & { a } \\ { a } & { 5 } & { - 1 } \end{array} \right)$$

Calculate Determinant

$$\left(a-7\right)\left(a+4\right)$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&1&-2\\3&1&a\\a&5&-1\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&1&-2&1&1\\3&1&a&3&1\\a&5&-1&a&5\end{matrix}\right)$$

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

$$-1+aa-2\times 3\times 5=a^{2}-31$$

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

$$a\left(-2\right)+5a-3=3a-3$$

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

$$a^{2}-31-\left(3a-3\right)$$

Subtract -3+3a from -31+a^{2}.

$$\left(a-7\right)\left(a+4\right)$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&1&-2\\3&1&a\\a&5&-1\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}1&a\\5&-1\end{matrix}\right))-det(\left(\begin{matrix}3&a\\a&-1\end{matrix}\right))-2det(\left(\begin{matrix}3&1\\a&5\end{matrix}\right))$$

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

$$-1-5a-\left(3\left(-1\right)-aa\right)-2\left(3\times 5-a\right)$$

Simplify.

$$-5a-1-\left(-a^{2}-3\right)-2\left(15-a\right)$$

Add the terms to obtain the final result.

$$\left(a-7\right)\left(a+4\right)$$

Evaluate

$$\left(\begin{matrix}1&1&-2\\3&1&a\\a&5&-1\end{matrix}\right)$$