Solve {lll}{4}&{8}&{2}{8}&{16}&{9}{13}&{26}&{7}

$$\left| \begin{array} { l l l } { 4 } & { 8 } & { 2 } \\ { 8 } & { 16 } & { 9 } \\ { 13 } & { 26 } & { 7 } \end{array} \right|$$

Evaluate

$$0$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}4&8&2\\8&16&9\\13&26&7\end{matrix}\right))$$

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

$$\left(\begin{matrix}4&8&2&4&8\\8&16&9&8&16\\13&26&7&13&26\end{matrix}\right)$$

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

$$4\times 16\times 7+8\times 9\times 13+2\times 8\times 26=1800$$

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

$$13\times 16\times 2+26\times 9\times 4+7\times 8\times 8=1800$$

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

$$1800-1800$$

Subtract 1800 from 1800.

$$0$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}4&8&2\\8&16&9\\13&26&7\end{matrix}\right))$$

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.

$$4det(\left(\begin{matrix}16&9\\26&7\end{matrix}\right))-8det(\left(\begin{matrix}8&9\\13&7\end{matrix}\right))+2det(\left(\begin{matrix}8&16\\13&26\end{matrix}\right))$$

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

$$4\left(16\times 7-26\times 9\right)-8\left(8\times 7-13\times 9\right)+2\left(8\times 26-13\times 16\right)$$

Simplify.

$$4\left(-122\right)-8\left(-61\right)$$

Add the terms to obtain the final result.

$$0$$

Factor

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