Question

Evaluate the determinant by

expanding by minors:

$\left|\begin{array}{ccc}3& -2& -2\\ 2& -1& 4\\ -1& 0& -3\end{array}\right|$

Short answer

The value of determinant is 7.

Step-by-step solution

Step 1. given Information.

Given square matrix:

$\left|\begin{array}{ccc}3& -2& -2\\ 2& -1& 4\\ -1& 0& -3\end{array}\right|$

Step 2. Write the concept.

To find determinant of $3\times 3$matrix using the minors we use the formula,

$\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=a_1\left|\begin{array}{cc}{b}_2& c_2\\ b_3& c_3\end{array}\right|-b_1\left|\begin{array}{cc}{a}_2& c_2\\ a_3& c_3\end{array}\right|+c_1\left|\begin{array}{cc}{a}_2& b_2\\ a_3& b_3\end{array}\right|$

expanding the minors can be done by selecting any row or column. Since 0 is present in second row and third column expanding by it will make calculation easier.

Step 3. Calculation.

Expand using third row.$$\begin{gathered} \left|\begin{array}{ccc}3& -2& -2\\ 2& -1& 4\\ -1& 0& -3\end{array}\right|=-1\left|\begin{array}{cc}-2& -2\\ -1& 4\end{array}\right|+0\left|\begin{array}{cc}3& -2\\ 2& 4\end{array}\right|-3\left|\begin{array}{cc}3& -2\\ 2& -1\end{array}\right| \\ =-1\left[\right(-2\left)\right(4)-(-2\left)\right(-1)-3[\left(3\right)(-1)-(-2)\left(2\right)] \\ =-1[-8-2]-3[-3+4] \\ =-1[-10]-3[1] \\ =10-3 \\ =7 \end{gathered}$$