Solve {ccc}{i}&{j}&{k}{0}&{-1}&{2}{2}&{-4}&{1}

$$\left| \begin{array} { c c c } { i } & { j } & { k } \\ { 0 } & { - 1 } & { 2 } \\ { 2 } & { - 4 } & { 1 } \end{array} \right|$$

Evaluate

$$4j+2k+7i$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}i&j&k\\0&-1&2\\2&-4&1\end{matrix}\right))$$

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

$$\left(\begin{matrix}i&j&k&i&j\\0&-1&2&0&-1\\2&-4&1&2&-4\end{matrix}\right)$$

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

$$-i+j\times 2\times 2=4j-i$$

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

$$2\left(-1\right)k-4\times \left(2i\right)=-8i-2k$$

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

$$4j-i-\left(-8i-2k\right)$$

Subtract -2k-8i from -i+4j.

$$4j+2k+7i$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}i&j&k\\0&-1&2\\2&-4&1\end{matrix}\right))$$

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.

$$idet(\left(\begin{matrix}-1&2\\-4&1\end{matrix}\right))-jdet(\left(\begin{matrix}0&2\\2&1\end{matrix}\right))+kdet(\left(\begin{matrix}0&-1\\2&-4\end{matrix}\right))$$

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

$$i\left(-1-\left(-4\times 2\right)\right)-j\left(-2\times 2\right)+k\left(-2\left(-1\right)\right)$$

Simplify.

$$7i-j\left(-4\right)+k\times 2$$

Add the terms to obtain the final result.

$$4j+2k+7i$$