Problem 1 - Precalculus

Question

Find the inverse of the matrix or state that the matrix is not invertible. $$ A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \( \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix} \).

Step 1: Review the Invertibility Condition

A 2x2 matrix is invertible if and only if its determinant is not zero. We first need to calculate the determinant of matrix \( A \).

Step 2: Calculate the Determinant

The determinant of matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) is calculated as: \[ ext{det}(A) = (1)(4) - (2)(3) = 4 - 6 = -2 \]Since the determinant is \(-2\), which is not zero, matrix \( A \) is invertible.

Step 3: Apply the Inverse Formula for a 2x2 Matrix

The formula for the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is: \[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]

Step 4: Substitute into the Formula

Using the matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \), substitute its values into the inverse formula:\[ A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \ 1.5 & -0.5 \end{bmatrix} \]

Key Concepts

Matrix Determinant

One of the key steps in finding the inverse of a matrix is calculating its determinant. The determinant is a special number that can be derived from a square matrix. It helps in understanding certain properties of the matrix, such as whether it is invertible or not.
The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula:

• (ad - bc): Multiply \( a \) by \( d \) and \( b \) by \( c \), then subtract the second result from the first.

In our example, the matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) has a determinant of \((1 \times 4) - (2 \times 3) = 4 - 6 = -2\).
This value is crucial as it confirms the potential to find an inverse.

Invertibility Condition

A matrix is considered invertible, or non-singular, if it meets specific conditions. The primary condition for a 2x2 matrix is that its determinant must not be zero. This is because the determinant being zero would imply that the matrix does not have a unique solution or cannot be inverted.

• The role of the determinant: If det(A) = 0, the matrix is singular and has no inverse. If det(A) ≠ 0, the matrix is non-singular and, thus, invertible.

In the original exercise, the determinant of matrix \( A \) was calculated to be -2. Because this is not zero, it assures us that matrix \( A \) is indeed invertible. Understanding this condition lays the foundation for knowing when we can appropriately proceed with finding an inverse.

2x2 Matrix Inverse

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in an identity matrix. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is found using a specific formula, provided that the determinant is non-zero.

• Inverse Formula: \[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]

Applying this to our matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \), with a determinant of -2, we have:

• \( A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \ 1.5 & -0.5 \end{bmatrix} \)

Thus, we have successfully found the inverse by substituting values into the formula, knowing our determinant ensured it was possible.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with the study of matrices and the rules for their manipulation. Understanding matrix operations is essential for studying linear transformations, solving systems of linear equations, and many more applications.

• Basic Operations: Includes addition, subtraction, and multiplication of matrices.
• Identity Matrix: A special type of matrix that acts like the number 1 in matrix multiplication. An identity matrix for a 2x2 is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).

In the context of the exercise, finding the inverse deals with more advanced concepts of matrix algebra—specifically, how to manipulate the elements within a matrix to achieve a specific result (identity matrix) when multiplying. This topic underscores the importance of understanding these operations and their results, elaborating on how they apply to tasks like finding inverses.