Solve {ccc}{-1}&{-2}&{-3}{-2}&{-3}&{-5}{-3}&{-4}&{-7}

$$\left| \begin{array} { c c c } { - 1 } & { - 2 } & { - 3 } \\ { - 2 } & { - 3 } & { - 5 } \\ { - 3 } & { - 4 } & { - 7 } \end{array} \right|$$

Evaluate

$$0$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-1&-2&-3\\-2&-3&-5\\-3&-4&-7\end{matrix}\right))$$

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

$$\left(\begin{matrix}-1&-2&-3&-1&-2\\-2&-3&-5&-2&-3\\-3&-4&-7&-3&-4\end{matrix}\right)$$

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

$$-\left(-3\right)\left(-7\right)-2\left(-5\right)\left(-3\right)-3\left(-2\right)\left(-4\right)=-75$$

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

$$-3\left(-3\right)\left(-3\right)-4\left(-5\right)\left(-1\right)-7\left(-2\right)\left(-2\right)=-75$$

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

$$-75-\left(-75\right)$$

Subtract -75 from -75.

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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).

$$det(\left(\begin{matrix}-1&-2&-3\\-2&-3&-5\\-3&-4&-7\end{matrix}\right))$$

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.

$$-det(\left(\begin{matrix}-3&-5\\-4&-7\end{matrix}\right))-\left(-2det(\left(\begin{matrix}-2&-5\\-3&-7\end{matrix}\right))\right)-3det(\left(\begin{matrix}-2&-3\\-3&-4\end{matrix}\right))$$

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

$$-\left(-3\left(-7\right)-\left(-4\left(-5\right)\right)\right)-\left(-2\left(-2\left(-7\right)-\left(-3\left(-5\right)\right)\right)\right)-3\left(-2\left(-4\right)-\left(-3\left(-3\right)\right)\right)$$

Simplify.

$$-1-\left(-2\left(-1\right)\right)-3\left(-1\right)$$

Add the terms to obtain the final result.

$$0$$

Factor

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