Solve {lll}{a}&{b}&{b}{b}&{a}&{b}{b}&{b}&{a}

$$\left( \begin{array} { l l l } { a } & { b } & { b } \\ { b } & { a } & { b } \\ { b } & { b } & { a } \end{array} \right)$$

Calculate Determinant

$$\left(a+2b\right)\left(a-b\right)^{2}$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}a&b&b\\b&a&b\\b&b&a\end{matrix}\right))$$

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

$$\left(\begin{matrix}a&b&b&a&b\\b&a&b&b&a\\b&b&a&b&b\end{matrix}\right)$$

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

$$aaa+bbb+bbb=a^{3}+2b^{3}$$

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

$$bab+bba+abb=3ab^{2}$$

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

$$a^{3}+2b^{3}-3ab^{2}$$

Subtract 3ab^{2} from 2b^{3}+a^{3}.

$$\left(a+2b\right)\left(a-b\right)^{2}$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}a&b&b\\b&a&b\\b&b&a\end{matrix}\right))$$

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.

$$adet(\left(\begin{matrix}a&b\\b&a\end{matrix}\right))-bdet(\left(\begin{matrix}b&b\\b&a\end{matrix}\right))+bdet(\left(\begin{matrix}b&a\\b&b\end{matrix}\right))$$

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

$$a\left(aa-bb\right)-b\left(ba-bb\right)+b\left(bb-ba\right)$$

Simplify.

$$a\left(a-b\right)\left(a+b\right)-bb\left(a-b\right)+bb\left(b-a\right)$$

Add the terms to obtain the final result.

$$\left(a+2b\right)\left(a-b\right)^{2}$$

Evaluate

$$\left(\begin{matrix}a&b&b\\b&a&b\\b&b&a\end{matrix}\right)$$