Solve {rrr}{-23}&{-9}&{6}{8}&{3}&{-2}{-3}&{1}&{1}

$$\left( \begin{array} { r r r } { - 23 } & { - 9 } & { 6 } \\ { 8 } & { 3 } & { - 2 } \\ { - 3 } & { 1 } & { 1 } \end{array} \right)$$

Calculate Determinant

$$5$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-23&-9&6\\8&3&-2\\-3&1&1\end{matrix}\right))$$

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

$$\left(\begin{matrix}-23&-9&6&-23&-9\\8&3&-2&8&3\\-3&1&1&-3&1\end{matrix}\right)$$

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

$$-23\times 3-9\left(-2\right)\left(-3\right)+6\times 8=-75$$

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

$$-3\times 3\times 6-2\left(-23\right)+8\left(-9\right)=-80$$

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

$$-75-\left(-80\right)$$

Subtract -80 from -75.

$$5$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-23&-9&6\\8&3&-2\\-3&1&1\end{matrix}\right))$$

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.

$$-23det(\left(\begin{matrix}3&-2\\1&1\end{matrix}\right))-\left(-9det(\left(\begin{matrix}8&-2\\-3&1\end{matrix}\right))\right)+6det(\left(\begin{matrix}8&3\\-3&1\end{matrix}\right))$$

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

$$-23\left(3-\left(-2\right)\right)-\left(-9\left(8-\left(-3\left(-2\right)\right)\right)\right)+6\left(8-\left(-3\times 3\right)\right)$$

Simplify.

$$-23\times 5-\left(-9\times 2\right)+6\times 17$$

Add the terms to obtain the final result.

$$5$$

Evaluate

$$\left(\begin{matrix}-23&-9&6\\8&3&-2\\-3&1&1\end{matrix}\right)$$