Problem 4 - Advanced Engineering Mathematics

Question

In Problems 1-20, fill in the blanks or answer true/false. $$ \text { If } \mathbf{A}=\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right), \text { then } \mathbf{A}^{-1}= $$ _________.

Step-by-Step Solution

Answer

The inverse matrix \( \mathbf{A}^{-1} \) is \( \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix} \).

Step 1: Recognize the Matrix Format

The given matrix \( \mathbf{A} \) is a 2x2 matrix with components \( \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} \). Our goal is to find the inverse of this matrix, denoted as \( \mathbf{A}^{-1} \).

Step 2: Use the Formula for the Inverse of a 2x2 Matrix

The general formula to find the inverse of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is \[ \mathbf{A}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]. For our matrix \( \mathbf{A} \), we have \( a = 1 \), \( b = 2 \), \( c = 3 \), and \( d = 4 \).

Step 3: Calculate the Determinant

The determinant \( ad-bc \) of matrix \( \mathbf{A} \) is \( (1)(4) - (2)(3) = 4 - 6 = -2 \). This determinant is needed for the inverse calculation.

Step 4: Apply the Formula

Substitute the values into the inverse matrix formula: \[ \mathbf{A}^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix} \].

Step 5: Simplify the Inverse Matrix

Multiply each element in the matrix by \( \frac{1}{-2} \), resulting in \( \mathbf{A}^{-1} = \begin{pmatrix} -2 & 1 \ 1.5 & -0.5 \end{pmatrix} \). This is the inverse of the original matrix \( \mathbf{A} \).

Key Concepts

Determinant

The determinant is a fundamental concept in linear algebra. It is a special number calculated from a square matrix. The determinant provides essential information about the matrix, such as whether it is invertible. For a 2x2 matrix, calculating the determinant is straightforward.

To find the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), use the formula:

• Calculate \( ad \), the product of the elements on the main diagonal.
• Calculate \( bc \), the product of the elements on the other diagonal.
• Subtract \( bc \) from \( ad \), i.e., \( ad - bc \).

In the given problem, our matrix \( \mathbf{A} \) has elements \( 1, 2, 3, \) and \( 4 \). The determinant is calculated as \( 1 \times 4 - 2 \times 3 = 4 - 6 = -2 \). This value is crucial for finding the inverse because a matrix with a determinant of zero does not have an inverse.

2x2 Matrix

A 2x2 matrix is one of the simplest types of matrices, consisting of two rows and two columns. It is a foundational structure in linear algebra, often used to represent transformations in a two-dimensional space.

In a general 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), each letter represents a number, and these values fill the four positions in the rows and columns. This matrix structure is useful because it is simple enough to understand but complex enough to model many real-world problems.

For example, a 2x2 matrix can represent:

• Coordinate transformations like rotations and scaling in geometry.
• Simple systems of linear equations.
• Network flows.

The matrix \( \mathbf{A} \) in our exercise is \( \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} \). This specific configuration allows us to explore matrix inversion, provided by the determinant is non-zero.

Inverse Matrix

An inverse matrix is like the 'opposite' of the original matrix in matrix multiplication. If you multiply a matrix and its inverse, you get the identity matrix, similar to how multiplying a number by its reciprocal gives 1. Finding the inverse of a matrix is a crucial operation in various mathematical and practical applications, such as solving systems of equations.

For a 2x2 matrix \( \mathbf{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the inverse \( \mathbf{A}^{-1} \) is found using the following formula:

• First, calculate the determinant \( ad - bc \).
• If the determinant is not zero, use it to find the inverse: \( \mathbf{A}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \).
• This formula rearranges and scales the original matrix elements according to the determinant value.

In our problem, after computing the determinant to be \(-2\), the inverse is found by scaling and rearranging \( \mathbf{A} \).
The inverse matrix we get here is \( \begin{pmatrix} -2 & 1 \ 1.5 & -0.5 \end{pmatrix} \), which, when multiplied by the original matrix, results in the identity matrix \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \). This confirms that the calculated inverse is correct.