Solve {l|c|c}hline9&{4}&{6}hline5&{7}&{7}hlineb&{8}&{6}hline

$$\left. \begin{array} { l | c | c } \hline 9 & { 4 } & { 6 } \\ \hline 5 & { 7 } & { 7 } \\ \hline b & { 8 } & { 6 } \\ \hline \end{array} \right.$$

Calculate Determinant

$$-14b-6$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}9&4&6\\5&7&7\\b&8&6\end{matrix}\right))$$

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

$$\left(\begin{matrix}9&4&6&9&4\\5&7&7&5&7\\b&8&6&b&8\end{matrix}\right)$$

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

$$9\times 7\times 6+4\times 7b+6\times 5\times 8=28b+618$$

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

$$b\times 7\times 6+8\times 7\times 9+6\times 5\times 4=42b+624$$

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

$$28b+618-\left(42b+624\right)$$

Subtract 42b+624 from 618+28b.

$$-14b-6$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}9&4&6\\5&7&7\\b&8&6\end{matrix}\right))$$

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.

$$9det(\left(\begin{matrix}7&7\\8&6\end{matrix}\right))-4det(\left(\begin{matrix}5&7\\b&6\end{matrix}\right))+6det(\left(\begin{matrix}5&7\\b&8\end{matrix}\right))$$

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

$$9\left(7\times 6-8\times 7\right)-4\left(5\times 6-b\times 7\right)+6\left(5\times 8-b\times 7\right)$$

Simplify.

$$9\left(-14\right)-4\left(30-7b\right)+6\left(40-7b\right)$$

Add the terms to obtain the final result.

$$-14b-6$$

Evaluate

$$\left(\begin{matrix}9&4&6\\5&7&7\\b&8&6\end{matrix}\right)$$