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

$$\left( \begin{array} { c c c } { 3 } & { - 1 } & { 3 } \\ { - 1 } & { 3 } & { - 1 } \\ { 3 } & { - 1 } & { 5 } \end{array} \right)$$

Calculate Determinant

$$16$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$3\times 3\times 5-\left(-3\right)+3\left(-1\right)\left(-1\right)=51$$

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

$$3\times 3\times 3-\left(-3\right)+5\left(-1\right)\left(-1\right)=35$$

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

$$51-35$$

Subtract 35 from 51.

$$16$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$3\times 14-\left(-\left(-2\right)\right)+3\left(-8\right)$$

Add the terms to obtain the final result.

$$16$$

Evaluate

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