Evaluate the determinant of the following matrix: | 3 -4 2 | | 1 5 -3 |

Understand the Problem

The question asks us to evaluate the determinant of a 3x3 matrix. We will use cofactor expansion to achieve this.

Answer

48

Steps to Solve

Write the matrix

The given matrix is: $$ \begin{vmatrix} 3 & -4 & 2 \ 1 & 5 & -3 \ -2 & 3 & 1 \end{vmatrix} $$

Expand along the first row

We will use the cofactor expansion along the first row. The determinant is given by: $$ \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} $$ Where $a_{ij}$ are the elements of the matrix and $C_{ij}$ are the corresponding cofactors.

Calculate the cofactors

$C_{11}$ is the determinant of the 2x2 matrix obtained by removing the first row and first column, multiplied by $(-1)^{1+1} = 1$: $$ C_{11} = \begin{vmatrix} 5 & -3 \ 3 & 1 \end{vmatrix} = (5)(1) - (-3)(3) = 5 + 9 = 14 $$

$C_{12}$ is the determinant of the 2x2 matrix obtained by removing the first row and second column, multiplied by $(-1)^{1+2} = -1$: $$ C_{12} = -\begin{vmatrix} 1 & -3 \ -2 & 1 \end{vmatrix} = -( (1)(1) - (-3)(-2) ) = -(1 - 6) = -(-5) = 5 $$

$C_{13}$ is the determinant of the 2x2 matrix obtained by removing the first row and third column, multiplied by $(-1)^{1+3} = 1$: $$ C_{13} = \begin{vmatrix} 1 & 5 \ -2 & 3 \end{vmatrix} = (1)(3) - (5)(-2) = 3 + 10 = 13 $$

Calculate the determinant

Now, substitute the values of $a_{11}, a_{12}, a_{13}$ and $C_{11}, C_{12}, C_{13}$ into the formula: $$ \det(A) = (3)(14) + (-4)(5) + (2)(13) = 42 - 20 + 26 = 48 $$

The determinant of the matrix is 48.

More Information

The determinant of a 3x3 matrix can be found using cofactor expansion along any row or column. The signs alternate in a checkerboard pattern when calculating the cofactors.

Tips

A common mistake is to forget the alternating signs when computing the cofactors. Another common mistake is to incorrectly calculate the 2x2 determinants.