Solve {ccc}{i}&{j}&{k}{4}&{-5}&{6}{-3}&{-5}&{9}

$$\left| \begin{array} { c c c } { i } & { j } & { k } \\ { 4 } & { - 5 } & { 6 } \\ { - 3 } & { - 5 } & { 9 } \end{array} \right|$$

Evaluate

$$-15i-35k-54j$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}i&j&k\\4&-5&6\\-3&-5&9\end{matrix}\right))$$

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

$$\left(\begin{matrix}i&j&k&i&j\\4&-5&6&4&-5\\-3&-5&9&-3&-5\end{matrix}\right)$$

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

$$-5i\times 9+j\times 6\left(-3\right)+k\times 4\left(-5\right)=-45i-20k-18j$$

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

$$-3\left(-5\right)k-5\times \left(6i\right)+9\times 4j=36j+15k-30i$$

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

$$-45i-20k-18j-\left(36j+15k-30i\right)$$

Subtract 15k-30i+36j from -45i-18j-20k.

$$-15i-35k-54j$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}i&j&k\\4&-5&6\\-3&-5&9\end{matrix}\right))$$

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.

$$idet(\left(\begin{matrix}-5&6\\-5&9\end{matrix}\right))-jdet(\left(\begin{matrix}4&6\\-3&9\end{matrix}\right))+kdet(\left(\begin{matrix}4&-5\\-3&-5\end{matrix}\right))$$

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

$$i\left(-5\times 9-\left(-5\times 6\right)\right)-j\left(4\times 9-\left(-3\times 6\right)\right)+k\left(4\left(-5\right)-\left(-3\left(-5\right)\right)\right)$$

Simplify.

$$-15i-j\times 54+k\left(-35\right)$$

Add the terms to obtain the final result.

$$-15i-35k-54j$$