Solve {ccc}{10}&{23}&{7}{15}&{4}&{22}{5}&{4}&{21}

$$\left( \begin{array} { c c c } { 10 } & { 23 } & { 7 } \\ { 15 } & { 4 } & { 22 } \\ { 5 } & { 4 } & { 21 } \end{array} \right)$$

Calculate Determinant

$$-4475$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}10&23&7\\15&4&22\\5&4&21\end{matrix}\right))$$

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

$$\left(\begin{matrix}10&23&7&10&23\\15&4&22&15&4\\5&4&21&5&4\end{matrix}\right)$$

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

$$10\times 4\times 21+23\times 22\times 5+7\times 15\times 4=3790$$

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

$$5\times 4\times 7+4\times 22\times 10+21\times 15\times 23=8265$$

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

$$3790-8265$$

Subtract 8265 from 3790.

$$-4475$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}10&23&7\\15&4&22\\5&4&21\end{matrix}\right))$$

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.

$$10det(\left(\begin{matrix}4&22\\4&21\end{matrix}\right))-23det(\left(\begin{matrix}15&22\\5&21\end{matrix}\right))+7det(\left(\begin{matrix}15&4\\5&4\end{matrix}\right))$$

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

$$10\left(4\times 21-4\times 22\right)-23\left(15\times 21-5\times 22\right)+7\left(15\times 4-5\times 4\right)$$

Simplify.

$$10\left(-4\right)-23\times 205+7\times 40$$

Add the terms to obtain the final result.

$$-4475$$

Evaluate

$$\left(\begin{matrix}10&23&7\\15&4&22\\5&4&21\end{matrix}\right)$$