Solve Matrix

$$\left| \begin{array} { l l } { 1 } & { 2 } & { 3 } \\ { 4 } & { 7 } & { 4 } \\ { 7 } & { 9 } & { 11 } \end{array} \right|$$

Evaluate

$$-30$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&2&3\\4&7&4\\7&9&11\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&2&3&1&2\\4&7&4&4&7\\7&9&11&7&9\end{matrix}\right)$$

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

$$7\times 11+2\times 4\times 7+3\times 4\times 9=241$$

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

$$7\times 7\times 3+9\times 4+11\times 4\times 2=271$$

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

$$241-271$$

Subtract 271 from 241.

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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).

$$det(\left(\begin{matrix}1&2&3\\4&7&4\\7&9&11\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}7&4\\9&11\end{matrix}\right))-2det(\left(\begin{matrix}4&4\\7&11\end{matrix}\right))+3det(\left(\begin{matrix}4&7\\7&9\end{matrix}\right))$$

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

$$7\times 11-9\times 4-2\left(4\times 11-7\times 4\right)+3\left(4\times 9-7\times 7\right)$$

Simplify.

$$41-2\times 16+3\left(-13\right)$$

Add the terms to obtain the final result.

$$-30$$

Factor

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