Solve {ccc}{hat{i}}&{hat{j}}&{hat{k}}{2}&{1}&{1}{1}&{-1}&{-3}

$$\left| \begin{array} { c c c } { \hat { i } } & { \hat { j } } & { \hat { k } } \\ { 2 } & { 1 } & { 1 } \\ { 1 } & { - 1 } & { - 3 } \end{array} \right|$$

Evaluate

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Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}0&0&0\\2&1&1\\1&-1&-3\end{matrix}\right))$$

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

$$\left(\begin{matrix}0&0&0&0&0\\2&1&1&2&1\\1&-1&-3&1&-1\end{matrix}\right)$$

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

$$\text{true}$$

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

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Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}0&0&0\\2&1&1\\1&-1&-3\end{matrix}\right))$$

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.

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Factor

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