Solve {ccc}{2}&{-3}&{-5}{1}&{9}&{-1}{5}&{-6}&{beta}

$$\left| \begin{array} { c c c } { 2 } & { - 3 } & { - 5 } \\ { 1 } & { 9 } & { - 1 } \\ { 5 } & { - 6 } & { \beta } \end{array} \right|$$

Evaluate

$$21\beta +258$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}2&-3&-5\\1&9&-1\\5&-6&\beta \end{matrix}\right))$$

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

$$\left(\begin{matrix}2&-3&-5&2&-3\\1&9&-1&1&9\\5&-6&\beta &5&-6\end{matrix}\right)$$

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

$$2\times 9\beta -3\left(-1\right)\times 5-5\left(-6\right)=18\beta +45$$

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

$$5\times 9\left(-5\right)-6\left(-1\right)\times 2+\beta \left(-3\right)=-3\beta -213$$

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

$$18\beta +45-\left(-3\beta -213\right)$$

Subtract -213-3\beta from 18\beta +45.

$$21\beta +258$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}2&-3&-5\\1&9&-1\\5&-6&\beta \end{matrix}\right))$$

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.

$$2det(\left(\begin{matrix}9&-1\\-6&\beta \end{matrix}\right))-\left(-3det(\left(\begin{matrix}1&-1\\5&\beta \end{matrix}\right))\right)-5det(\left(\begin{matrix}1&9\\5&-6\end{matrix}\right))$$

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

$$2\left(9\beta -\left(-6\left(-1\right)\right)\right)-\left(-3\left(\beta -5\left(-1\right)\right)\right)-5\left(-6-5\times 9\right)$$

Simplify.

$$2\left(9\beta -6\right)-\left(-3\left(\beta +5\right)\right)-5\left(-51\right)$$

Add the terms to obtain the final result.

$$21\beta +258$$

Integrate w.r.t. β

$$\frac{21\beta ^{2}}{2}+258\beta +С$$