Problem 8 - Precalculus: Mathematical for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 2.2 & -1.4 \\ 0.5 & 1.0 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 2.9.

Step 1: Determine the Type of Matrix

Recognize that the given matrix is a 2x2 matrix. A determinant can be calculated for square matrices, so we can proceed with finding the determinant.

Step 2: Identify Matrix Elements

Label the elements of the matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Here, \( a = 2.2 \), \( b = -1.4 \), \( c = 0.5 \), \( d = 1.0 \).

Step 3: Apply the Determinant Formula for a 2x2 Matrix

Use the formula for the determinant of a 2x2 matrix, \( ext{det}(A) = ad - bc \). Substituting the values, we get \( ext{det}(A) = (2.2)(1.0) - (-1.4)(0.5) \).

Step 4: Perform the Multiplication

Calculate each part of the formula: \( 2.2 \times 1.0 = 2.2 \) and \( -1.4 \times 0.5 = -0.7 \).

Step 5: Subtract the Products to Find the Determinant

Substitute the results from Step 4 into the formula: \( 2.2 - (-0.7) = 2.2 + 0.7 = 2.9 \). This is the determinant of the matrix.

Key Concepts

2x2 Matrix

A 2x2 matrix is a square matrix, meaning it has the same number of rows and columns. Specifically, it has two rows and two columns. Each element in a 2x2 matrix is usually denoted by the variables

• \(a\)
• \(b\)
• \(c\)
• \(d\)

placed in the format: \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]
The structure of a 2x2 matrix allows for easy computations and is often the first type of matrix studied when learning linear algebra. Understanding the configuration of a matrix is crucial when performing operations like calculating its determinant or engaging in matrix multiplication.

Determinant Formula

The determinant of a 2x2 matrix is a specific numeric value that provides important information about the matrix. The calculation is straightforward, using the formula:
\[\text{det}(A) = ad - bc\]where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix. This formula derives from the need to determine if a matrix is invertible, among other uses. A foundational understanding of the determinant allows for deeper insight into the properties of the matrix, such as whether the matrix has an inverse. In our case, we calculated with the elements:

• \(a = 2.2\)
• \(b = -1.4\)
• \(c = 0.5\)
• \(d = 1.0\)

By substituting these values into the formula, we arrived at a determinant of 2.9, indicating the calculations involved multiplication and subtraction as detailed in the previous solution.

Matrix Multiplication

Matrix multiplication is a fundamental operation not only limited to forming new matrices, but also to processes like finding determinants. While a determinant relies on multiplication only within its formula, understanding matrix multiplication itself is vital to broader applications.
In matrix multiplication, the dimensions of the matrices must align properly for multiplication to be successful. This means if you multiply a matrix \(A\) (of dimensions 2x2) with another matrix \(B\) (of dimensions 2x1), the resulting matrix will be of dimensions 2x1.

• Each element of the resulting matrix is a sum of products of elements from the rows of the first matrix and the columns of the second matrix.
• Just as in our determinant operation, the individual elements were multiplied (like \(a \, \text{and} \, d\)) before performing additional operations such as subtraction.

Grasping these concepts allows further exploration into diverse fields where matrices are pivotal, such as computer graphics and systems of linear equations.