Solve {lll}{y}&{0}&{u}{y}&{a}&{n}{g}&{x}&{y}

$$\left| \begin{array} { l l l } { y } & { 0 } & { u } \\ { y } & { a } & { n } \\ { g } & { x } & { y } \end{array} \right|$$

Evaluate

$$uxy-nxy+ay^{2}-agu$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}y&0&u\\y&a&n\\g&x&y\end{matrix}\right))$$

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

$$\left(\begin{matrix}y&0&u&y&0\\y&a&n&y&a\\g&x&y&g&x\end{matrix}\right)$$

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

$$yay+uyx=y\left(ux+ay\right)$$

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

$$gau+xny=nxy+agu$$

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

$$y\left(ux+ay\right)-\left(nxy+agu\right)$$

Subtract gau+xny from y\left(ay+ux\right).

$$y\left(ux-nx+ay\right)-agu$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}y&0&u\\y&a&n\\g&x&y\end{matrix}\right))$$

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.

$$ydet(\left(\begin{matrix}a&n\\x&y\end{matrix}\right))+udet(\left(\begin{matrix}y&a\\g&x\end{matrix}\right))$$

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

$$y\left(ay-xn\right)+u\left(yx-ga\right)$$

Simplify.

$$y\left(ay-nx\right)+u\left(xy-ag\right)$$

Add the terms to obtain the final result.

$$u\left(xy-ag\right)+y\left(ay-nx\right)$$

Integrate w.r.t. x

$$\frac{uyx^{2}}{2}-\frac{nyx^{2}}{2}-agux+axy^{2}+С$$