Solve {ccc}{-7}&{7}&{9}{2}&{-1}&{-5}{7}&{-7}&{-9}

$$\left| \begin{array} { c c c } { - 7 } & { 7 } & { 9 } \\ { 2 } & { - 1 } & { - 5 } \\ { 7 } & { - 7 } & { - 9 } \end{array} \right|$$

Evaluate

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Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-7&7&9\\2&-1&-5\\7&-7&-9\end{matrix}\right))$$

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

$$\left(\begin{matrix}-7&7&9&-7&7\\2&-1&-5&2&-1\\7&-7&-9&7&-7\end{matrix}\right)$$

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

$$-7\left(-1\right)\left(-9\right)+7\left(-5\right)\times 7+9\times 2\left(-7\right)=-434$$

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

$$7\left(-1\right)\times 9-7\left(-5\right)\left(-7\right)-9\times 2\times 7=-434$$

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

$$-434-\left(-434\right)$$

Subtract -434 from -434.

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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).

$$det(\left(\begin{matrix}-7&7&9\\2&-1&-5\\7&-7&-9\end{matrix}\right))$$

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.

$$-7det(\left(\begin{matrix}-1&-5\\-7&-9\end{matrix}\right))-7det(\left(\begin{matrix}2&-5\\7&-9\end{matrix}\right))+9det(\left(\begin{matrix}2&-1\\7&-7\end{matrix}\right))$$

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

$$-7\left(-\left(-9\right)-\left(-7\left(-5\right)\right)\right)-7\left(2\left(-9\right)-7\left(-5\right)\right)+9\left(2\left(-7\right)-7\left(-1\right)\right)$$

Simplify.

$$-7\left(-26\right)-7\times 17+9\left(-7\right)$$

Add the terms to obtain the final result.

$$0$$

Factor

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