Solve {ccc}{2-99}&{6}&{-24}{6}&{2-130}&{30}{24}&{30}&{2-71}=

$$\left| \begin{array} { c c c } { 2 - 99 } & { 6 } & { - 24 } \\ { 6 } & { 2 - 130 } & { 30 } \\ { 24 } & { 30 } & { 2 - 71 } \end{array} \right| =$$

Evaluate

$$-840648$$

Steps Using Sarrus' Rule

Subtract 99 from 2 to get -97.

$$det(\left(\begin{matrix}-97&6&-24\\6&2-130&30\\24&30&2-71\end{matrix}\right))$$

Subtract 130 from 2 to get -128.

$$det(\left(\begin{matrix}-97&6&-24\\6&-128&30\\24&30&2-71\end{matrix}\right))$$

Subtract 71 from 2 to get -69.

$$det(\left(\begin{matrix}-97&6&-24\\6&-128&30\\24&30&-69\end{matrix}\right))$$

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

$$\left(\begin{matrix}-97&6&-24&-97&6\\6&-128&30&6&-128\\24&30&-69&24&30\end{matrix}\right)$$

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

$$-97\left(-128\right)\left(-69\right)+6\times 30\times 24-24\times 6\times 30=-856704$$

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

$$24\left(-128\right)\left(-24\right)+30\times 30\left(-97\right)-69\times 6\times 6=-16056$$

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

$$-856704-\left(-16056\right)$$

Subtract -16056 from -856704.

$$-840648$$

Steps Using Expansion by Minors

Subtract 99 from 2 to get -97.

$$det(\left(\begin{matrix}-97&6&-24\\6&2-130&30\\24&30&2-71\end{matrix}\right))$$

Subtract 130 from 2 to get -128.

$$det(\left(\begin{matrix}-97&6&-24\\6&-128&30\\24&30&2-71\end{matrix}\right))$$

Subtract 71 from 2 to get -69.

$$det(\left(\begin{matrix}-97&6&-24\\6&-128&30\\24&30&-69\end{matrix}\right))$$

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.

$$-97det(\left(\begin{matrix}-128&30\\30&-69\end{matrix}\right))-6det(\left(\begin{matrix}6&30\\24&-69\end{matrix}\right))-24det(\left(\begin{matrix}6&-128\\24&30\end{matrix}\right))$$

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

$$-97\left(-128\left(-69\right)-30\times 30\right)-6\left(6\left(-69\right)-24\times 30\right)-24\left(6\times 30-24\left(-128\right)\right)$$

Simplify.

$$-97\times 7932-6\left(-1134\right)-24\times 3252$$

Add the terms to obtain the final result.

$$-840648$$

Factor

$$-840648$$