Evaluate the determinant of the following matrix: | 3 -4 2 | | 1 5 -3 | |-2 3 1 |
Understand the Problem
The question is asking us to evaluate the determinant of the given 3x3 matrix. We will use the cofactor expansion method to calculate the determinant. First choose a row or column to expand along, then calculate the determinant of the 2x2 submatrices for each element in the chosen row or column, paying attention to the sign based on the position of the element.
Answer
48
Answer for screen readers
48
Steps to Solve
- Write out the matrix
The given matrix is: $$ \begin{vmatrix} 3 & -4 & 2 \ 1 & 5 & -3 \ -2 & 3 & 1 \end{vmatrix} $$
- Choose a row or column to expand along
Let's expand along the first row.
- Calculate the determinant using cofactor expansion
The determinant is calculated as follows: $$ \begin{aligned} \text{det} &= 3 \cdot \begin{vmatrix} 5 & -3 \ 3 & 1 \end{vmatrix} - (-4) \cdot \begin{vmatrix} 1 & -3 \ -2 & 1 \end{vmatrix} + 2 \cdot \begin{vmatrix} 1 & 5 \ -2 & 3 \end{vmatrix} \ &= 3 \cdot (5 \cdot 1 - (-3) \cdot 3) + 4 \cdot (1 \cdot 1 - (-3) \cdot (-2)) + 2 \cdot (1 \cdot 3 - 5 \cdot (-2)) \ &= 3 \cdot (5 + 9) + 4 \cdot (1 - 6) + 2 \cdot (3 + 10) \ &= 3 \cdot 14 + 4 \cdot (-5) + 2 \cdot 13 \ &= 42 - 20 + 26 \ &= 48 \end{aligned} $$
48
More Information
The determinant of the matrix is 48. The cofactor expansion method allows us to calculate the determinant of larger matrices by breaking them down into smaller 2x2 matrices.
Tips
A common mistake is to forget the alternating signs when using cofactor expansion. Remember the checkerboard pattern of signs: $$ \begin{vmatrix}
- & - & + \
- & + & - \
- & - & + \end{vmatrix} $$ Another common mistake involves incorrectly computing the determinant of the 2x2 submatrices. Ensure you multiply correctly and subtract in the correct order.
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