Problem 53 - College Algebra

Question

Explain how to evaluate a third-order determinant.

Step-by-Step Solution

Answer

The determinant of a 3x3 matrix \[A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}\] can be calculated using the formula \[|A| = a(ei - fh) - b(di - fg) + c(dh - eg)\]. Replace each variable in the formula with the corresponding element from the matrix, compute the result to find the determinant.

Step 1: Identify the 3x3 matrix

Write down your 3x3 matrix. It should contain nine elements arranged in three rows and three columns. It can be denoted as \[A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}\]

Step 2: Expansion by Minors

Expand the determinant across a row or a column. The expansion is generally done across the top row but it can be done using any row or column. This gives the formula for the determinant as \[|A| = a(ei - fh) - b(di - fg) + c(dh - eg)\]

Step 3: Calculate the Result

Simply substitute the values from your matrix into the formula. Evaluate the expressions in parentheses first, according to the order of operations, then multiply and add/subtract as indicated to find the determinant of the 3x3 matrix.