Problem 55 - College Algebra

Question

Explain how to evaluate a second-order determinant.

Step-by-Step Solution

Answer

A second-order determinant is evaluated by multiplying the elements along the main diagonal and subtracting the product of the remaining two elements. For example, the determinant of the matrix \( \[ \begin{pmatrix} a & b \ c & d \end{pmatrix}\] \) is \(ad - bc\).

Step 1: Understand the Structure of a Second-Order Determinant

A 2x2 or second-order determinant is a square matrix with two rows and two columns. It is represented as \( \[ \begin{pmatrix} a & b \ c & d \end{pmatrix}\] \). The letters a, b, c and d represent numbers in the matrix.

Step 2: Apply the Formula for Evaluating a Second-Order Determinant

The formula for evaluating a second-order determinant is \(ad - bc\). It means multiplying the elements along the main diagonal ('a' and 'd') and subtracting the product of the other two numbers ('b' and 'c').

Step 3: Compute the Value

After applying the formula, calculate the value. For example, \( \[ \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}\] \) gives \( (1*4) - (2*3) = 4 - 6 = -2 \).