Problem 12 - Algebra for College Students

Question

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{ll} -2 & 0 \\ -3 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The multiplicative inverse is \( \begin{bmatrix} -\frac{1}{2} & 0 \\ -\frac{3}{10} & \frac{1}{5} \end{bmatrix} \).

Step 1: Identify the matrix elements

The given matrix is \( \begin{bmatrix} -2 & 0 \ -3 & 5 \end{bmatrix} \). Let's denote the elements as \( a = -2 \), \( b = 0 \), \( c = -3 \), and \( d = 5 \). The matrix is a 2x2 matrix.

Step 2: Calculate the determinant

The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( ad - bc \). Substituting the values, we get the determinant: \(-2 \cdot 5 - 0 \cdot -3 = -10 - 0 = -10\).

Step 3: Check if the determinant is zero

The determinant is \(-10\). Since this is not zero, the matrix does have an inverse.

Step 4: Apply the formula for the inverse of a 2x2 matrix

The formula for the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Substitute in the values: \( \frac{1}{-10} \begin{bmatrix} 5 & 0 \ 3 & -2 \end{bmatrix} \).

Step 5: Simplify the result

Multiply each element in the matrix by \( \frac{1}{-10} \): \( \begin{bmatrix} -\frac{1}{2} & 0 \ -\frac{3}{10} & \frac{1}{5} \end{bmatrix} \).

Step 6: Verify the inverse (Optional)

To confirm, multiply the original matrix by the calculated inverse and check if it equals the identity matrix. \( \begin{bmatrix} -2 & 0 \ -3 & 5 \end{bmatrix} \times \begin{bmatrix} -\frac{1}{2} & 0 \ -\frac{3}{10} & \frac{1}{5} \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), which confirms our inverse is correct.

Key Concepts

Determinant

The determinant is a key concept in linear algebra that helps us understand many properties of matrices. For a 2x2 matrix, the determinant can determine if a matrix has an inverse. If the determinant is zero, the matrix does not have an inverse. This is because the inverse relies on dividing by the determinant, and division by zero is undefined.
For a 2x2 matrix represented as \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]the determinant is calculated using the formula \( ad - bc \).

• In our exercise, with matrix \[ \begin{bmatrix} -2 & 0 \ -3 & 5 \end{bmatrix} \],the determinant is \( -2 \times 5 - 0 \times (-3) = -10 \).
• This non-zero result confirms an inverse does exist.

Understanding the determinant is crucial when working with inverses and other operations involving matrices.

2x2 Matrix

The 2x2 matrix is one of the simplest forms of matrices and is frequently used in linear algebra due to its simplicity and relevance. It consists of two rows and two columns, making it compact yet powerful enough for various calculations and transformations.
A typical 2x2 matrix is written as:\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]where:

• \(a\), \(b\), \(c\), and \(d\) are elements of the matrix.
• These elements can be any real numbers or even complex numbers, depending on the context.

For operations like finding the inverse, 2x2 matrices are particularly convenient. The rules and formulas are more straightforward compared to larger matrices.

Identity Matrix

In matrix algebra, the identity matrix plays a critical role similar to the number 1 in arithmetic. When a matrix is multiplied by the identity matrix, it remains unchanged. This feature is essential when verifying whether a matrix has been correctly inverted.
For a 2x2 identity matrix:\[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]

• If \(A\) is any 2x2 matrix, then \(A \times I = I \times A = A\).
• Using the identity matrix, the correctness of an inverse is confirmed. If multiplying the original matrix and its inverse yields the identity matrix, the inversion is correct.

Thus, every matrix operation involving an inverse checks against the identity matrix to ensure validity.

Matrices Multiplication

Matrix multiplication is a fundamental operation in linear algebra and is crucial when dealing with matrices and their transformations. Multiplying matrices involves combining rows from the first matrix with columns of the second matrix. The process is not commutative, meaning the order of multiplication matters.
For any two 2x2 matrices \(A\) and \(B\):

• The element in the first row first column of the result comes from multiplying elements of the first row of \(A\) by those in the first column of \(B\) and summing the results.
• Similarly, the rest of the resulting matrix can be filled by following this pattern for the respective rows and columns.

When verifying the inverse, we multiply the given matrix by its supposed inverse. If the product is the identity matrix, our inverse calculation is validated. This procedure confirms the inverse matrix's precision by returning the identity matrix when correct.