Problem 13 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rr}{0.4} & {-1.2} \\ {0.3} & {0.6}\end{array}\right]\)

Step-by-Step Solution

Answer

The inverse matrix is \(\begin{bmatrix} 1 & 2 \\ -0.5 & 0.6667 \end{bmatrix}\).

Step 1: Check if the determinant is zero

To find the inverse of a matrix, we first need to check if the determinant is non-zero. The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is calculated as \(ad - bc\). For our matrix: \(\begin{bmatrix} 0.4 & -1.2 \ 0.3 & 0.6 \end{bmatrix}\), the determinant is \((0.4 \times 0.6) - (-1.2 \times 0.3)\). Calculate this value.

Step 2: Calculate the determinant

Compute the determinant: \((0.4 \times 0.6) - (-1.2 \times 0.3) = 0.24 + 0.36 = 0.6\). The determinant is \(0.6\), which is not zero, so the matrix is invertible.

Step 3: Use the formula for the inverse

Since the matrix is invertible, we can find its inverse using the formula for a 2x2 matrix's inverse: \(\frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). Substitute \(a = 0.4\), \(b = -1.2\), \(c = 0.3\), and \(d = 0.6\) into this formula.

Step 4: Substitute and calculate the inverse matrix

Apply the values into the inverse formula: \(\frac{1}{0.6}\begin{bmatrix} 0.6 & 1.2 \ -0.3 & 0.4 \end{bmatrix}\). Multiply \(1/0.6\) into each entry of the matrix to get the inverse: \(\begin{bmatrix} 1 & 2 \ -0.5 & 0.6667 \end{bmatrix}\).

Step 5: Verify the inverse matrix

To ensure correctness, verify by multiplying the original and inverse matrices, \(\begin{bmatrix} 0.4 & -1.2 \ 0.3 & 0.6 \end{bmatrix}\begin{bmatrix} 1 & 2 \ -0.5 & 0.6667 \end{bmatrix}\), and vice versa. Both products should result in the identity matrix \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that helps determine if a matrix has an inverse. It's a crucial value when dealing with square matrices. For a 2x2 matrix, the determinant is calculated using the formula:

• For matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is \(ad - bc\).

In the original exercise, the matrix \(\begin{bmatrix} 0.4 & -1.2 \ 0.3 & 0.6 \end{bmatrix}\) yielded a determinant of \(0.6\). This number was obtained by calculating \((0.4 \times 0.6) - (-1.2 \times 0.3) = 0.24 + 0.36 = 0.6\).
The determinant is not zero (unlike matrices with a determinant of zero, which cannot be inverted). This allowed us to proceed to find the inverse of the matrix.

2x2 Matrix

A 2x2 matrix is a simple, square matrix consisting of two rows and two columns. These matrices are quite common in math problems because their operations are straightforward.

• Each element in the matrix is typically represented by variables like \(a, b, c,\) and \(d\).
• They are often used to explore matrix concepts because they are less complex than larger matrices, making calculations like finding the determinant or inverse more manageable.

For example, the matrix in the given exercise \(\begin{bmatrix} 0.4 & -1.2 \ 0.3 & 0.6 \end{bmatrix}\) shows a practical 2x2 form where you can easily apply mathematical operations such as finding the determinant or the inverse. Learning with 2x2 matrices builds a foundation for understanding larger matrices.

Inverse Matrix Formula

To find the inverse of a 2x2 matrix, you need to check if the determinant is not zero. If the determinant is zero, the matrix does not have an inverse. For our non-zero determinant, the inverse formula comes into play:

• The inverse of \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by:\[\frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]

This formula rearranges the matrix while scaling it by the reciprocal of the determinant. In the exercise, plugging in the numbers,

• \(a = 0.4\), \(b = -1.2\), \(c = 0.3\), \(d = 0.6\), resulted in\[\frac{1}{0.6}\begin{bmatrix} 0.6 & 1.2 \ -0.3 & 0.4 \end{bmatrix} = \begin{bmatrix} 1 & 2 \ -0.5 & 0.6667 \end{bmatrix}\]

This inverse matrix can be used to reverse any operation applied by the original matrix.

Identity Matrix

The identity matrix is a key concept in understanding matrix inversion. It acts as the "1" of matrix multiplication, meaning any matrix multiplied by the identity matrix remains unchanged. For a 2x2 matrix, the identity matrix is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

• The verification step in matrix inversion involves confirming that the product of the original matrix and its inverse results in the identity matrix.
• In the exercise, multiplying the original matrix \(\begin{bmatrix} 0.4 & -1.2 \ 0.3 & 0.6 \end{bmatrix}\) with its calculated inverse \(\begin{bmatrix} 1 & 2 \ -0.5 & 0.6667 \end{bmatrix}\) yields \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\), confirming the validity of the inverse.

This property is essential for representing the idea of "undoing" a matrix operation, ensuring that solving matrix equations is consistent with operations similar to algebraic equations.