Solve {ccc}{7}&{-3}&{2}{-3}&{7}&{-1}{-2}&{-1}&{5}

$$\left| \begin{array} { c c c } { 7 } & { - 3 } & { 2 } \\ { - 3 } & { 7 } & { - 1 } \\ { - 2 } & { - 1 } & { 5 } \end{array} \right|$$

Evaluate

$$221$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}7&-3&2\\-3&7&-1\\-2&-1&5\end{matrix}\right))$$

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

$$\left(\begin{matrix}7&-3&2&7&-3\\-3&7&-1&-3&7\\-2&-1&5&-2&-1\end{matrix}\right)$$

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

$$7\times 7\times 5-3\left(-1\right)\left(-2\right)+2\left(-3\right)\left(-1\right)=245$$

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

$$-2\times 7\times 2-\left(-7\right)+5\left(-3\right)\left(-3\right)=24$$

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

$$245-24$$

Subtract 24 from 245.

$$221$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}7&-3&2\\-3&7&-1\\-2&-1&5\end{matrix}\right))$$

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.

$$7det(\left(\begin{matrix}7&-1\\-1&5\end{matrix}\right))-\left(-3det(\left(\begin{matrix}-3&-1\\-2&5\end{matrix}\right))\right)+2det(\left(\begin{matrix}-3&7\\-2&-1\end{matrix}\right))$$

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

$$7\left(7\times 5-\left(-\left(-1\right)\right)\right)-\left(-3\left(-3\times 5-\left(-2\left(-1\right)\right)\right)\right)+2\left(-3\left(-1\right)-\left(-2\times 7\right)\right)$$

Simplify.

$$7\times 34-\left(-3\left(-17\right)\right)+2\times 17$$

Add the terms to obtain the final result.

$$221$$

Factor

$$13\times 17$$