Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} -1 & 2 \\ 3 & -4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -2.

Step 1: Identify the Matrix Elements

The given matrix is a 2x2 matrix: \(\begin{bmatrix} -1 & 2 \3 & -4\end{bmatrix}.\) Let's label the elements of the matrix as follows:- \(a = -1\)- \(b = 2\)- \(c = 3\)- \(d = -4\)

Step 2: Apply the Determinant Formula for a 2x2 Matrix

The formula for the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \c & d\end{bmatrix}\) is given by: \[ad - bc\] Substituting the identified values into this formula, we get: \((-1)\cdot(-4) - (3)\cdot(2)\)

Step 3: Perform the Calculations

Now calculate each part of the determinant formula:- The product of \(a\) and \(d\):\((-1)\cdot(-4) = 4\)- The product of \(b\) and \(c\):\(2\cdot3 = 6\) Substitute these products back into the formula: \(4 - 6\)

Step 4: Determine the Final Value

By completing the subtraction, we find:\(4 - 6 = -2\). Therefore, the determinant of the given matrix is \(-2\).

Key Concepts

Matrix Elements

A matrix is a rectangular array of numbers arranged in rows and columns. Each individual number within the matrix is referred to as an element. For a 2x2 matrix, which has two rows and two columns, there are four elements. These elements are typically labeled as follows:

• The element in the first row and first column is denoted as \(a\).
• The element in the first row and second column is \(b\).
• The element in the second row and first column is \(c\).
• The element in the second row and second column is \(d\).

Understanding these labels helps in performing matrix operations such as calculating the determinant. It's essential to correctly identify these elements to avoid errors in calculations.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices and is defined by its two rows and two columns. It can be represented by:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]This type of matrix is widely used in basic linear algebra because of its simplicity. Each 2x2 matrix contains four elements. This simple structure is helpful for introductory calculations, particularly when determining its determinant, which is a measure of its size or scale.

Matrix Operations

Matrix operations involve a variety of mathematical procedures that can be performed on matrices, such as addition, subtraction, multiplication, and finding the determinant. Here, we focus on finding the determinant of a 2x2 matrix, a specific type of matrix operation.The formula for the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is:\[ad - bc\]This involves multiplying the elements diagonally and then subtracting the products. Specifically, you multiply the top-left element \(a\) by the bottom-right element \(d\), then subtract the product of the top-right element \(b\) and the bottom-left element \(c\). For our example, that would be:

• \((-1) \times (-4) = 4\)
• \((2) \times (3) = 6\)
• Then subtract: \(4 - 6 = -2\)

This simple subtraction gives us the determinant of the matrix. Understanding how to perform this operation is fundamental in linear algebra for solving systems of equations and other applications.