Problem 43 - College Algebra

Question

Find \(A^{-1}\) and check. $$A=\left[\begin{array}{rr}e^{x} & e^{3 x} \\\\-e^{3 x} & e^{5 x}\end{array}\right]$$

Step-by-Step Solution

Answer

The matrix A does not have an inverse.

Step 1: Compute the Determinant

The first step is to calculate the determinant of the given matrix A. The determinant is given by the formula \( ad - bc \), where a, b, c and d are elements of the 2x2 matrix. The determinant of Matrix A therefore is: Det(A) = \( (e^x)(e^{5x}) - ((e^{3x})(-e^{3x})) = (e^{6x} - e^{6x}) = 0. \)

Step 2: Check if the Determinant is zero

The next step is to check if the determinant is 0. If the determinant is 0, then the matrix does not have an inverse. In our case, the determinant is 0 and thus it can be concluded that the Matrix A does not have an inverse. In very specific cases where the determinant of a matrix is 0, the matrix could still be invertible but this would be the exception not the rule.

Key Concepts

Determinant Calculation

When working with matrices, one of the fundamental concepts to grasp is how to find the determinant. The determinant is a special number that can tell us a lot about the matrix. For a 2x2 matrix like the one in our example, the formula to compute the determinant is \(ad - bc\). Here, \(a, b, c,\) and \(d\) are the elements of the matrix arranged as follows:

• \(a\) is the element at the first row and first column
• \(b\) is the element at the first row and second column
• \(c\) is the element at the second row and first column
• \(d\) is the element at the second row and second column

In our provided matrix, after substitution, the determinant resulted in 0, which is a key indication that the matrix does not have an inverse.

2x2 Matrix

A 2x2 matrix is a simple matrix that consists of two rows and two columns. These matrices are easier to handle compared to larger matrices and form the basic building blocks for understanding matrix operations.
In our example, the 2x2 matrix is represented as:\[A = \begin{bmatrix} e^x & e^{3x} \ -e^{3x} & e^{5x} \end{bmatrix}\]Each element in the matrix can include variables and exponents, as in our case with exponential functions.
Understanding the layout of a 2x2 matrix helps us utilize formulas like the determinant mathematical operation effectively, facilitating further operations like matrix inversion when possible.

Inverse Matrix

The concept of an inverse matrix is essential when solving systems of equations. The inverse of a matrix, denoted as \(A^{-1}\), is a matrix that, when multiplied by the original matrix \(A\), yields the identity matrix.
However, not all matrices have an inverse. For a 2x2 matrix, the presence of an inverse depends on the determinant being non-zero. If the determinant is zero, as is the case in our example, the inverse does not exist.
This concept is crucial in understanding when and how a matrix can be used to transform data or solve equations, emphasizing the importance of first checking the determinant.