Solve {rrr}{-3}&{4}&{16}{5}&{5}&{13}{-4}&{3}&{12}

$$\left| \begin{array} { r r r } { - 3 } & { 4 } & { 16 } \\ { 5 } & { 5 } & { 13 } \\ { - 4 } & { 3 } & { 12 } \end{array} \right|$$

Evaluate

$$49$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-3&4&16\\5&5&13\\-4&3&12\end{matrix}\right))$$

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

$$\left(\begin{matrix}-3&4&16&-3&4\\5&5&13&5&5\\-4&3&12&-4&3\end{matrix}\right)$$

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

$$-3\times 5\times 12+4\times 13\left(-4\right)+16\times 5\times 3=-148$$

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

$$-4\times 5\times 16+3\times 13\left(-3\right)+12\times 5\times 4=-197$$

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

$$-148-\left(-197\right)$$

Subtract -197 from -148.

$$49$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-3&4&16\\5&5&13\\-4&3&12\end{matrix}\right))$$

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.

$$-3det(\left(\begin{matrix}5&13\\3&12\end{matrix}\right))-4det(\left(\begin{matrix}5&13\\-4&12\end{matrix}\right))+16det(\left(\begin{matrix}5&5\\-4&3\end{matrix}\right))$$

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

$$-3\left(5\times 12-3\times 13\right)-4\left(5\times 12-\left(-4\times 13\right)\right)+16\left(5\times 3-\left(-4\times 5\right)\right)$$

Simplify.

$$-3\times 21-4\times 112+16\times 35$$

Add the terms to obtain the final result.

$$49$$

Factor

$$7^{2}$$