Problem 51 - College Algebra

Question

Explain how to evaluate a second-order determinant.

Step-by-Step Solution

Answer

The determinant of a 2x2 matrix \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \] is calculated by subtracting the product of the off-diagonal elements from the product of the main diagonal elements and is given by the formula: \( det(A) = ad - bc \)

Step 1: Identify the Matrix Elements

Consider the 2x2 matrix as follows:\[ \begin{pmatrix}a & b \c & d\end{pmatrix}\]Where \(a\), \(b\), \(c\) and \(d\) are elements of the matrix.

Step 2: Multiply the Main Diagonal Elements

The main diagonal is from top left to bottom right. In this case, it consists of elements \(a\) and \(d\). Multiply these elements together to get \(ad\).

Step 3: Multiply the Other Diagonal Elements

The other diagonal runs from top right to bottom left, consisting of elements \(b\) and \(c\). Multiply these two elements to get \(bc\).

Step 4: Evaluate the Determinant

Subtract the product from Step 3 from the product from Step 2. The formula for the determinant of a 2x2 matrix is given by:\[det(A) = ad - bc\]Therefore, applying this formula, one gets the determinant of the given matrix.