Solve {lll}{1}&{1}&{0}{4}&{-5}&{1}{3}&{1}&{1}

$$\left| \begin{array} { l l l } { 1 } & { 1 } & { 0 } \\ { 4 } & { - 5 } & { 1 } \\ { 3 } & { 1 } & { 1 } \end{array} \right|$$

Evaluate

$$-7$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&1&0\\4&-5&1\\3&1&1\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&1&0&1&1\\4&-5&1&4&-5\\3&1&1&3&1\end{matrix}\right)$$

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

$$-5+3=-2$$

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

$$1+4=5$$

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

$$-2-5$$

Subtract 5 from -2.

$$-7$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&1&0\\4&-5&1\\3&1&1\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}-5&1\\1&1\end{matrix}\right))-det(\left(\begin{matrix}4&1\\3&1\end{matrix}\right))$$

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

$$-5-1-\left(4-3\right)$$

Simplify.

$$-6-1$$

Add the terms to obtain the final result.

$$-7$$

Factor

$$-7$$