Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rr} 2 & -4 \\ 1 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix is \(\begin{bmatrix} 0.3 & 0.4 \\ -0.1 & 0.2 \end{bmatrix}\).

Step 1: Determine the Matrix A

The given matrix is \( A = \begin{bmatrix} 2 & -4 \ 1 & 3 \end{bmatrix} \).

Step 2: Calculate the Determinant of A

For a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is given by \(ad - bc\). Thus for \(A\), the determinant is \(2 \times 3 - (-4) \times 1 = 6 + 4 = 10\).

Step 3: Check if the Inverse Exists

A matrix has an inverse if and only if its determinant is not zero. The determinant of \(A\) is 10, which is not zero. Thus, \(A\) has an inverse.

Step 4: Find the Adjugate of A

The adjugate (adjoint) of \(A\) in a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). Hence, for \(A\), the adjugate is \(\begin{bmatrix} 3 & 4 \ -1 & 2 \end{bmatrix}\).

Step 5: Calculate the Inverse of A

The inverse of \(A\) is given by \(\frac{1}{\text{det}(A)} \times \text{adj}(A)\). Since \(\text{det}(A) = 10\), the inverse \(A^{-1}\) is \(\frac{1}{10} \times \begin{bmatrix} 3 & 4 \ -1 & 2 \end{bmatrix} = \begin{bmatrix} \frac{3}{10} & \frac{4}{10} \ \frac{-1}{10} & \frac{2}{10} \end{bmatrix}\) or simplifed, \(\begin{bmatrix} 0.3 & 0.4 \ -0.1 & 0.2 \end{bmatrix} \).

Key Concepts

Determinant Calculation

When dealing with matrices, particularly when finding the inverse, calculating the determinant is crucial. For a 2x2 matrix of the form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is a simple yet powerful tool. It is calculated using the formula:

• \( ad - bc \).

This expression looks at the product of the diagonal elements subtracted by the product of the off-diagonal elements.
For example, consider the matrix \( A = \begin{bmatrix} 2 & -4 \ 1 & 3 \end{bmatrix} \). Here, the determinant becomes:

• \( 2 \times 3 - (-4) \times 1 = 6 + 4 = 10 \).

Calculating the determinant informs you if the matrix can have an inverse, as a non-zero determinant implies the matrix is invertible.

Adjugate Matrix

An adjugate matrix, also known as the adjoint, plays a key role in finding the inverse of a matrix. For a 2x2 matrix, this is formed by swapping the diagonal elements, and changing the signs of the off-diagonal elements.

Given a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), its adjugate is:

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

Let's take the matrix \( A = \begin{bmatrix} 2 & -4 \ 1 & 3 \end{bmatrix} \). The adjugate of this matrix becomes:

• \( \begin{bmatrix} 3 & 4 \ -1 & 2 \end{bmatrix} \).

Thus, understanding and computing the adjugate matrix is a necessary step to finding the inverse of a matrix.

Inverse Matrix Existence

The existence of an inverse matrix is contingent upon one critical property: a non-zero determinant. A matrix can only have an inverse if its determinant is not equal to zero. For every square matrix that meets this criterion, there exists another matrix such that when multiplied together, they result in the identity matrix.
In our example matrix \( A = \begin{bmatrix} 2 & -4 \ 1 & 3 \end{bmatrix} \), we calculated the determinant as 10, which is not zero, affirming that \( A \) has an inverse. This means we can proceed to find \( A^{-1} \) using its determinant and adjugate. The process of confirming that an inverse exists is a critical checkpoint in matrix algebra.

2x2 Matrices

A 2x2 matrix is one of the simplest forms of matrices and is characterized by having two rows and two columns. This compact structure makes calculations like determinants and inverses straightforward and manageable.
In 2x2 matrices:

• The determinant formula \( ad - bc \) is quick to compute.
• The inverse, when it exists, can be calculated using the adjugate matrix and the determinant through the formula \[ A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A) \].

Because these matrices involve simpler calculations and fewer elements, they are ideal for illustrating foundational matrix concepts. Mastering the operations on 2x2 matrices paves the way for understanding more complex matrix operations in higher dimensions.