Solve {ccc}{9}&{12}&{15}{19}&{26}&{33}{29}&{40}&{51}

$$\left( \begin{array} { c c c } { 9 } & { 12 } & { 15 } \\ { 19 } & { 26 } & { 33 } \\ { 29 } & { 40 } & { 51 } \end{array} \right)$$

Calculate Determinant

$$0$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}9&12&15\\19&26&33\\29&40&51\end{matrix}\right))$$

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

$$\left(\begin{matrix}9&12&15&9&12\\19&26&33&19&26\\29&40&51&29&40\end{matrix}\right)$$

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

$$9\times 26\times 51+12\times 33\times 29+15\times 19\times 40=34818$$

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

$$29\times 26\times 15+40\times 33\times 9+51\times 19\times 12=34818$$

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

$$34818-34818$$

Subtract 34818 from 34818.

$$0$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}9&12&15\\19&26&33\\29&40&51\end{matrix}\right))$$

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.

$$9det(\left(\begin{matrix}26&33\\40&51\end{matrix}\right))-12det(\left(\begin{matrix}19&33\\29&51\end{matrix}\right))+15det(\left(\begin{matrix}19&26\\29&40\end{matrix}\right))$$

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

$$9\left(26\times 51-40\times 33\right)-12\left(19\times 51-29\times 33\right)+15\left(19\times 40-29\times 26\right)$$

Simplify.

$$9\times 6-12\times 12+15\times 6$$

Add the terms to obtain the final result.

$$0$$

Evaluate

$$\left(\begin{matrix}9&12&15\\19&26&33\\29&40&51\end{matrix}\right)$$