Problem 57 - Algebra and Trigonometry

Question

Explain how to evaluate a third-order determinant.

Step-by-Step Solution

Answer

To evaluate a third-order determinant, set up a 3x3 matrix, identify the first row elements, and create a 2x2 'minor' matrix for each first row element by eliminating their associated row and column. Then compute the determinant of each 'minor' matrix and finally calculate the third-order determinant by combining these computations with attention to alternating signs.

Step 1: Setup the 3x3 Matrix

First, a 3x3 matrix must be set up using given values or coefficients. This matrix will be your determinant. We'll use, for this example, a matrix of the form \[A=\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\]

Step 2: Identify First Row Elements

Identify the elements in the first row: a, b, and c. These three elements will each have a 'minor' matrix associated with them. The process subsequently is to multiply the element by the determinant of its minor matrix.

Step 3: Find the 'Minor' Matrices

Create a 2x2 'minor' matrix for each first row element by removing their associated row and column. For the first element \(a\), remove the first column and row to get: \[\begin{bmatrix}e & f \\ h & i \end{bmatrix}\] Repeat this for each element in the first row.

Step 4: Calculate Determinants of 'Minor' Matrices

Find the determinant of each 'minor' matrix. This is done by multiplying diagonally and subtracting the two results. For the first minor, \[\text{det}=\begin{vmatrix}e & f \\ h & i \end{vmatrix}=(e*i)-(f*h)\] Similarly, find the determinants for the other two 'minor' matrices.

Step 5: Compute the Third-Order Determinant

Finally, calculate the third-order determinant by following the formula \(a*\text{det1}-b*\text{det2}+c*\text{det3}\). This involves multiplication of each first row element of the determinant by the determinant of its corresponding minor matrix. Consider the signs + - + in front of each cofactor expansion.