Evaluate the determinant of the following matrix: [[3, -4, 2], [1, 5, -3], [-2, 3, 1]]

Steps to Solve

1. Write out the matrix

First, let's write out the matrix for which we need to find the determinant: $$ \begin{bmatrix} 3 & -4 & 2 \ 1 & 5 & -3 \ -2 & 3 & 1 \end{bmatrix} $$

2. Calculate the determinant using cofactor expansion along the first row

We can use the cofactor expansion method along the first row to calculate the determinant. This involves multiplying each element in the first row by its corresponding cofactor and then summing these products. The formula is: $$ \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} $$ Here, $a_{ij}$ represents the element in the $i$-th row and $j$-th column, and $C_{ij}$ is the cofactor of that element.

3. Calculate the cofactors

The cofactor $C_{ij}$ is calculated as $(-1)^{i+j}M_{ij}$, where $M_{ij}$ is the minor of the element $a_{ij}$. The minor is the determinant of the submatrix formed by removing the $i$-th row and $j$-th column.

4. Calculate $C_{11}$ $C_{11} = (-1)^{1+1} \cdot \det \begin{bmatrix} 5 & -3 \ 3 & 1 \end{bmatrix} = (1) \cdot (5\cdot1 - (-3)\cdot3) = 5 + 9 = 14$

5. Calculate $C_{12}$ $C_{12} = (-1)^{1+2} \cdot \det \begin{bmatrix} 1 & -3 \ -2 & 1 \end{bmatrix} = (-1) \cdot (1\cdot1 - (-3)\cdot(-2)) = -(1 - 6) = -(-5) = 5$

6. Calculate $C_{13}$ $C_{13} = (-1)^{1+3} \cdot \det \begin{bmatrix} 1 & 5 \ -2 & 3 \end{bmatrix} = (1) \cdot (1\cdot3 - 5\cdot(-2)) = 3 + 10 = 13$

7. Substitute the values into the determinant formula

Now substitute the values of the elements and cofactors into the determinant formula:

$\det(A) = 3 \cdot C_{11} + (-4) \cdot C_{12} + 2 \cdot C_{13} = 3(14) + (-4)(5) + 2(13) = 42 - 20 + 26 = 48$