Solve {ccc}{a}&{2}&{3}{1}&{-4}&{-51}{-1}&{3}&{38}

$$\left( \begin{array} { c c c } { a } & { 2 } & { 3 } \\ { 1 } & { - 4 } & { - 51 } \\ { - 1 } & { 3 } & { 38 } \end{array} \right)$$

Calculate Determinant

$$a+23$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}a&2&3\\1&-4&-51\\-1&3&38\end{matrix}\right))$$

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

$$\left(\begin{matrix}a&2&3&a&2\\1&-4&-51&1&-4\\-1&3&38&-1&3\end{matrix}\right)$$

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

$$a\left(-4\right)\times 38+2\left(-51\right)\left(-1\right)+3\times 3=111-152a$$

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

$$-\left(-4\right)\times 3+3\left(-51\right)a+38\times 2=88-153a$$

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

$$111-152a-\left(88-153a\right)$$

Subtract 88-153a from -152a+111.

$$a+23$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}a&2&3\\1&-4&-51\\-1&3&38\end{matrix}\right))$$

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.

$$adet(\left(\begin{matrix}-4&-51\\3&38\end{matrix}\right))-2det(\left(\begin{matrix}1&-51\\-1&38\end{matrix}\right))+3det(\left(\begin{matrix}1&-4\\-1&3\end{matrix}\right))$$

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

$$a\left(-4\times 38-3\left(-51\right)\right)-2\left(38-\left(-\left(-51\right)\right)\right)+3\left(3-\left(-\left(-4\right)\right)\right)$$

Simplify.

$$a-2\left(-13\right)+3\left(-1\right)$$

Add the terms to obtain the final result.

$$a+23$$

Evaluate

$$\left(\begin{matrix}a&2&3\\1&-4&-51\\-1&3&38\end{matrix}\right)$$