Solve {lll}{i}&{j}&{k}{p}&{q}&{r}{x}&{y}&{2}

$$\left| \begin{array} { l l l } { i } & { j } & { k } \\ { p } & { q } & { r } \\ { x } & { y } & { 2 } \end{array} \right|$$

Evaluate

$$jrx-kqx+kpy-iry-2jp+2iq$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}i&j&k\\p&q&r\\x&y&2\end{matrix}\right))$$

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

$$\left(\begin{matrix}i&j&k&i&j\\p&q&r&p&q\\x&y&2&x&y\end{matrix}\right)$$

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

$$iq\times 2+jrx+kpy=jrx+kpy+2iq$$

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

$$xqk+yri+2pj=kqx+iry+2jp$$

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

$$jrx+kpy+2iq-\left(kqx+iry+2jp\right)$$

Subtract xqk+iyr+2pj from 2iq+jrx+kpy.

$$jrx-kqx+kpy-iry-2jp+2iq$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}i&j&k\\p&q&r\\x&y&2\end{matrix}\right))$$

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.

$$idet(\left(\begin{matrix}q&r\\y&2\end{matrix}\right))-jdet(\left(\begin{matrix}p&r\\x&2\end{matrix}\right))+kdet(\left(\begin{matrix}p&q\\x&y\end{matrix}\right))$$

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

$$i\left(q\times 2-yr\right)-j\left(p\times 2-xr\right)+k\left(py-xq\right)$$

Simplify.

$$i\left(2q-ry\right)-j\left(2p-rx\right)+k\left(py-qx\right)$$

Add the terms to obtain the final result.

$$jrx-kqx+kpy-iry-2jp+2iq$$