Solve {lll}{4}&{54}&{3}{9}&{16}&{23}{78}&{14}&{7}

$$\left| \begin{array} { l l l } { 4 } & { 54 } & { 3 } \\ { 9 } & { 16 } & { 23 } \\ { 78 } & { 14 } & { 7 } \end{array} \right|$$

Evaluate

$$89268$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}4&54&3\\9&16&23\\78&14&7\end{matrix}\right))$$

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

$$\left(\begin{matrix}4&54&3&4&54\\9&16&23&9&16\\78&14&7&78&14\end{matrix}\right)$$

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

$$4\times 16\times 7+54\times 23\times 78+3\times 9\times 14=97702$$

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

$$78\times 16\times 3+14\times 23\times 4+7\times 9\times 54=8434$$

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

$$97702-8434$$

Subtract 8434 from 97702.

$$89268$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}4&54&3\\9&16&23\\78&14&7\end{matrix}\right))$$

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.

$$4det(\left(\begin{matrix}16&23\\14&7\end{matrix}\right))-54det(\left(\begin{matrix}9&23\\78&7\end{matrix}\right))+3det(\left(\begin{matrix}9&16\\78&14\end{matrix}\right))$$

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

$$4\left(16\times 7-14\times 23\right)-54\left(9\times 7-78\times 23\right)+3\left(9\times 14-78\times 16\right)$$

Simplify.

$$4\left(-210\right)-54\left(-1731\right)+3\left(-1122\right)$$

Add the terms to obtain the final result.

$$89268$$

Factor

$$2^{2}\times 3\times 43\times 173$$