Solve {lll}{i}&{j}&{k}{1}&{2}&{3}{4}&{5}&{6}

$$\left| \begin{array} { l l l } { i } & { j } & { k } \\ { 1 } & { 2 } & { 3 } \\ { 4 } & { 5 } & { 6 } \end{array} \right|$$

Evaluate

$$6j-3k-3i$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}i&j&k\\1&2&3\\4&5&6\end{matrix}\right))$$

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

$$\left(\begin{matrix}i&j&k&i&j\\1&2&3&1&2\\4&5&6&4&5\end{matrix}\right)$$

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

$$2i\times 6+j\times 3\times 4+k\times 5=12j+5k+12i$$

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

$$4\times 2k+5\times \left(3i\right)+6j=6j+8k+15i$$

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

$$12j+5k+12i-\left(6j+8k+15i\right)$$

Subtract 8k+15i+6j from 12i+12j+5k.

$$6j-3k-3i$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}i&j&k\\1&2&3\\4&5&6\end{matrix}\right))$$

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.

$$idet(\left(\begin{matrix}2&3\\5&6\end{matrix}\right))-jdet(\left(\begin{matrix}1&3\\4&6\end{matrix}\right))+kdet(\left(\begin{matrix}1&2\\4&5\end{matrix}\right))$$

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

$$i\left(2\times 6-5\times 3\right)-j\left(6-4\times 3\right)+k\left(5-4\times 2\right)$$

Simplify.

$$-3i-j\left(-6\right)+k\left(-3\right)$$

Add the terms to obtain the final result.

$$6j-3k-3i$$