Solve {ccc}{3}&{-2}&{-1}{0}&{-1}&{1}{-3}&{7}&{-1}

$$\left( \begin{array} { c c c } { 3 } & { - 2 } & { - 1 } \\ { 0 } & { - 1 } & { 1 } \\ { - 3 } & { 7 } & { - 1 } \end{array} \right)$$

Calculate Determinant

$$-9$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}3&-2&-1\\0&-1&1\\-3&7&-1\end{matrix}\right))$$

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

$$\left(\begin{matrix}3&-2&-1&3&-2\\0&-1&1&0&-1\\-3&7&-1&-3&7\end{matrix}\right)$$

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

$$3\left(-1\right)\left(-1\right)-2\left(-3\right)=9$$

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

$$-3\left(-1\right)\left(-1\right)+7\times 3=18$$

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

$$9-18$$

Subtract 18 from 9.

$$-9$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}3&-2&-1\\0&-1&1\\-3&7&-1\end{matrix}\right))$$

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.

$$3det(\left(\begin{matrix}-1&1\\7&-1\end{matrix}\right))-\left(-2det(\left(\begin{matrix}0&1\\-3&-1\end{matrix}\right))\right)-det(\left(\begin{matrix}0&-1\\-3&7\end{matrix}\right))$$

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

$$3\left(-\left(-1\right)-7\right)-\left(-2\left(-\left(-3\right)\right)\right)-\left(-\left(-3\left(-1\right)\right)\right)$$

Simplify.

$$3\left(-6\right)-\left(-2\times 3\right)-\left(-3\right)$$

Add the terms to obtain the final result.

$$-9$$

Evaluate

$$\left(\begin{matrix}3&-2&-1\\0&-1&1\\-3&7&-1\end{matrix}\right)$$