Evaluate the determinant of the following third order matrix: | 3 -4 2 | | 1 5 -3 | | -2 3 1 |
Understand the Problem
The question asks to evaluate the determinant of the provided 3x3 matrix, and will require a step-by-step solution.
Answer
$48$
Answer for screen readers
$48$
Steps to Solve
- Write down the formula for the determinant of a 3x3 matrix
Given a 3x3 matrix:
$$ \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} $$
The determinant is calculated as:
$a(ei - fh) - b(di - fg) + c(dh - eg)$
- Identify the elements of the matrix
From the given matrix:
$$ \begin{vmatrix} 3 & -4 & 2 \ 1 & 5 & -3 \ -2 & 3 & 1 \end{vmatrix} $$
We have: $a = 3$, $b = -4$, $c = 2$, $d = 1$, $e = 5$, $f = -3$, $g = -2$, $h = 3$, $i = 1$
- Substitute the values into the determinant formula
Substitute these values into the determinant formula:
$3(51 - (-3)3) - (-4)(11 - (-3)(-2)) + 2(13 - 5(-2))$
- Calculate the determinant
Simplify the expression:
$3(5 + 9) + 4(1 - 6) + 2(3 + 10)$ $3(14) + 4(-5) + 2(13)$ $42 - 20 + 26$ $48$
$48$
More Information
The determinant of the matrix is 48. Determinants are used in various applications, such as solving systems of linear equations, finding eigenvalues, and calculating the area or volume of geometric figures.
Tips
A common mistake is making errors in the arithmetic when calculating the cofactors and expanding along a row or column. Pay close attention to signs and multiplication. Another common mistake is not following the correct formula for calculating the determinant, especially with the signs. It's important to remember the alternating + and - signs in the expansion.
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