Problem 29 - College Algebra with Modeling and Visualization

Question

LetA be the given matrix. Find \(A^{-1}\). $$ \left[\begin{array}{ll} 0.5 & -1.5 \\ 0.2 & -0.5 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the given matrix is \( \begin{bmatrix} -10 & 30 \\ -4 & 10 \end{bmatrix} \).

Step 1: Verify Matrix Size for Inversion

Check if the given matrix \( A \) is a square matrix. A matrix is invertible only if it is square. The given matrix is 2x2, thus it can potentially be inverted.

Step 2: Calculate the Determinant

The determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( \text{det}(A) = ad - bc \). For our matrix: \( a = 0.5 \), \( b = -1.5 \), \( c = 0.2 \), \( d = -0.5 \). Thus, \( \text{det}(A) = (0.5)(-0.5) - (-1.5)(0.2) = -0.25 + 0.3 = 0.05 \).

Step 3: Check if the Determinant is Non-Zero

The determinant must be non-zero for the matrix to be invertible. Here, \( \text{det}(A) = 0.05 \), which is non-zero. Hence, the inverse exists.

Step 4: Use the Formula for the Inverse of a 2x2 Matrix

The inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Substituting the given matrix values: \( A^{-1} = \frac{1}{0.05} \begin{bmatrix} -0.5 & 1.5 \ -0.2 & 0.5 \end{bmatrix} = 20 \begin{bmatrix} -0.5 & 1.5 \ -0.2 & 0.5 \end{bmatrix} \).

Step 5: Simplify and Write the Inverse Matrix

Multiply the matrix by 20: \( A^{-1} = \begin{bmatrix} (-0.5) \times 20 & 1.5 \times 20 \ -0.2 \times 20 & 0.5 \times 20 \end{bmatrix} = \begin{bmatrix} -10 & 30 \ -4 & 10 \end{bmatrix} \).

Key Concepts

Determinant of a Matrix

Understanding the determinant of a matrix is crucial, especially when determining whether a matrix is invertible. In simple terms, the determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula for the determinant is:
\[ \text{det}(A) = ad - bc \]
To put it into perspective, the determinant gives us vital insights into the matrix's properties. For instance:

• A determinant of zero means the matrix is non-invertible, or singular, indicating no unique solution exists for the system of equations it represents.
• A non-zero determinant confirms the matrix is invertible, meaning there is a unique solution.

Applying this to our example, calculating the determinant \( (0.5 \times -0.5) - (-1.5 \times 0.2) = -0.25 + 0.3 = 0.05 \) is a critical step showing that \( A^{-1} \) is possible because 0.05 is not zero, confirming invertibility.

2x2 Matrix

A 2x2 matrix is a compact structure consisting of two rows and two columns. Mathematically, it is represented as:
\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]
Here, \( a, b, c, \) and \( d \) are the elements of the matrix. 2x2 matrices are often the starting point for learning about matrices because they include enough complexity to learn key concepts while remaining manageable.
In practical applications, 2x2 matrices can represent transformations in 2D space, such as rotations, scalings, and shears. These transformations are crucial in fields like computer graphics, physics, and engineering.
Understanding the properties of 2x2 matrices, such as their determinant and inverse, allows us to perform operations like solving systems of linear equations, among other tasks in matrix algebra.

Inverse of a Matrix

The inverse of a matrix is a crucial concept in algebra. It is essentially the "undo" transformation of a matrix. For a 2x2 matrix \( A \), the inverse, noted as \( A^{-1} \), is the matrix that when multiplied by \( A \) results in the identity matrix:
\[A \times A^{-1} = I\]
where \( I \) is the identity matrix with ones on the diagonal and zeros elsewhere. To find this inverse, especially for a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula is:
\[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]
Importantly, this formula only works if the matrix is invertible, which implies the determinant \( ad-bc \) is non-zero. Calculating the inverse is a straightforward process, once your determinant check is complete. With our example, the operations resulted in finding \( A^{-1} = \begin{bmatrix} -10 & 30 \ -4 & 10 \end{bmatrix} \). Each step confirms the matrix's invertibility and provides the tools for various mathematical applications.

Matrix Algebra

Matrix algebra encompasses a wide range of operations that involve matrices, like addition, multiplication, determinant calculation, and inversion. It's a field that transcends into numerous applications in science, engineering, and computer science.
For matrix operations, some fundamental rules include:

• Matrix addition and subtraction require matrices of the same dimensions.
• Matrix multiplication involves an intricate process of dot products and requires matching inner dimensions.
• Special properties, like commutativity (can’t switch matrices in multiplication), define operability.

In our given problem, matrix inversion represents an essential part of matrix algebra. After verifying that a matrix is square and determining its determinant, knowing how to find the inverse allows for tasks such as solving linear systems.
Mastering these concepts equips students with the necessary mathematical tools to build and explore more complex operations and systems, making matrix algebra a foundational element in both academic study and professional fields.