Solve {ccc}{i}&{j}&{k}{-18}&{0}&{10}{9}&{5}&{-5}

$$\left| \begin{array} { c c c } { i } & { j } & { k } \\ { - 18 } & { 0 } & { 10 } \\ { 9 } & { 5 } & { - 5 } \end{array} \right|$$

Evaluate

$$-50i-90k$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}i&j&k\\-18&0&10\\9&5&-5\end{matrix}\right))$$

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

$$\left(\begin{matrix}i&j&k&i&j\\-18&0&10&-18&0\\9&5&-5&9&5\end{matrix}\right)$$

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

$$j\times 10\times 9+k\left(-18\right)\times 5=90j-90k$$

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

$$5\times \left(10i\right)-5\left(-18\right)j=90j+50i$$

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

$$90j-90k-\left(90j+50i\right)$$

Subtract 50i+90j from 90j-90k.

$$-50i-90k$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}i&j&k\\-18&0&10\\9&5&-5\end{matrix}\right))$$

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.

$$idet(\left(\begin{matrix}0&10\\5&-5\end{matrix}\right))-jdet(\left(\begin{matrix}-18&10\\9&-5\end{matrix}\right))+kdet(\left(\begin{matrix}-18&0\\9&5\end{matrix}\right))$$

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

$$i\left(-5\times 10\right)-j\left(-18\left(-5\right)-9\times 10\right)+k\left(-18\right)\times 5$$

Simplify.

$$-50i+k\left(-90\right)$$

Add the terms to obtain the final result.

$$-50i-90k$$