Problem 13 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} -6 & 4 \\ -3 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix \(A\) is not invertible because its determinant is zero.

Step 1: Determine if the Matrix is Invertible

To find the inverse of the matrix \(A\), we first need to check if it is invertible. A matrix is invertible if its determinant is non-zero. Compute the determinant of \(A\) using the formula: \[ \text{det}(A) = ad - bc \]for a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\). In our case, \(a = -6\), \(b = 4\), \(c = -3\), and \(d = 2\). Thus,\[ \text{det}(A) = (-6)(2) - (4)(-3) = -12 + 12 = 0 \].

Step 2: Conclude about Invertibility

Since the determinant of matrix \(A\) is 0, \(A\) is not invertible. A matrix with a determinant of zero does not have an inverse.

Key Concepts

Determinant

When you're dealing with matrices, one important concept to grasp is the determinant. Especially in the context of finding the inverse of a matrix. The determinant is a special number calculated from the elements of a square matrix. In the case of a 2x2 matrix, it provides vital information about the matrix's properties. Given a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is found using the formula:

• \( \text{det}(A) = ad - bc \)

This simple calculation can determine whether a matrix can have an inverse. Generally, if the determinant is zero, the matrix doesn't have an inverse. This property is crucial because it means the matrix cannot be used for certain mathematical operations like solving linear equations.

2x2 Matrix

The 2x2 matrix is a fundamental building block in linear algebra and appears frequently in various mathematical contexts. A 2x2 matrix has two rows and two columns, which makes it the simplest non-trivial example of a square matrix. For any given entries \( a, b, c, \) and \( d \), you can express a 2x2 matrix as:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

This format is especially handy for transformations and switching between coordinate systems. Many operations, like finding the determinant or inverse, are well defined and computationally straightforward for 2x2 matrices. Understanding them is a solid foundation for progressing into higher dimensions like 3x3 matrices, where operations become slightly more complex but follow similar principles.

Invertibility

Invertibility is a core concept in linear algebra, referring to a matrix's ability to have an inverse. Not all matrices can be inverted—that is, some don't possess an inverse at all. A matrix has an inverse if and only if its determinant is non-zero. This is especially simple to see in a 2x2 matrix scenario. For a given matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), if \( ad - bc e 0 \), the matrix is invertible.

• The inverse of the matrix \( A \), represented as \( A^{-1} \), can theoretically be calculated if it is invertible.
• If the determinant is zero, the matrix is not invertible, and no such inverse \( A^{-1} \) exists.

Understanding invertibility helps in many mathematical applications, such as solving systems of linear equations, analyzing vectors, and more. Being familiar with how to determine invertibility also aids in developing a strong foundational understanding of matrix operations.