Problem 16 - Algebra for College Students

Question

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rr} -3 & 4 \\ 1 & -2 \end{array}\right] $$

Step-by-Step Solution

Answer

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

Step 1: Verify the Matrix is 2x2

We need to find the multiplicative inverse of the given 2x2 matrix:\[\begin{bmatrix} -3 & 4 \ 1 & -2 \end{bmatrix}\] The matrix is indeed 2x2, so we can proceed.

Step 2: Calculate the Determinant

The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by \(ad-bc\). For our matrix, we have:\[\text{det}(A) = (-3)(-2) - (4)(1) = 6 - 4 = 2\]Since the determinant is not zero, an inverse exists.

Step 3: Use the Formula for the Inverse of a 2x2 Matrix

The inverse of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by:\[\frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\] Applying this to our matrix:\[A^{-1} = \frac{1}{2} \begin{bmatrix} -2 & -4 \ -1 & -3 \end{bmatrix}\] Simplifying gives:\[A^{-1} = \begin{bmatrix} -1 & -2 \ -0.5 & -1.5 \end{bmatrix}\]

Step 4: Verify the Inverse Calculation (Optional Step)

To verify our inverse calculation, we can multiply the original matrix by its inverse and check if it yields the identity matrix. Perform matrix multiplication:\[\begin{bmatrix} -3 & 4 \ 1 & -2 \end{bmatrix} \begin{bmatrix} -1 & -2 \ -0.5 & -1.5 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]Both elements on the main diagonal are 1, and the off-diagonal elements are 0, confirming our calculations.

Key Concepts

Determinant of a Matrix

A determinant is a special number that you can calculate from a square matrix. It gives us important information about the matrix, such as whether an inverse exists. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant can be found by the formula:

• \(ad - bc\)

This value is crucial. If the determinant is zero, it means the matrix does not have an inverse.When calculating the determinant for a matrix like \(\begin{bmatrix} -3 & 4 \ 1 & -2 \end{bmatrix}\), you calculate:

• \((-3) \times (-2) - 4 \times 1 = 6 - 4 = 2\)

Since the result is not zero, the matrix has an inverse. Remember, the determinant is like the key to unlocking whether you can find that elusive inverse!

2x2 Matrix

A 2x2 matrix is a simple yet powerful grid that consists of two rows and two columns. This type of matrix is denoted as: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\] Here, \(a, b, c,\) and \(d\) are elements that make up the matrix.These elements could be any number. Understanding the layout of a 2x2 matrix simplifies many calculations, especially when finding determinants or inverses.

These matrices possess unique characteristics:

• They have straightforward operations, such as addition, multiplication, and inversion.
• Their determinants are easily calculated, providing a swift pathway to finding inverses.

Operating on a 2x2 matrix is one of the first steps in matrix algebra and lays the foundation for more complex matrices.

Matrix Multiplication

Matrix multiplication is a way to combine two matrices to produce another matrix. In the realm of 2x2 matrices, it's often used to check the accuracy of an inverse calculation.To multiply matrices, follow these basic steps:

• Take each element from a row in the first matrix.
• Multiply and sum them with the corresponding elements from a column of the second matrix.

For instance, to verify that our matrix and its inverse produce the identity matrix, you multiply\[\begin{bmatrix} -3 & 4 \ 1 & -2 \end{bmatrix}\]by\[\begin{bmatrix} -1 & -2 \ -0.5 & -1.5 \end{bmatrix}\].The result should be\[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\].

Key points when performing matrix multiplication:

• The number of columns in the first matrix should equal the number of rows in the second matrix.
• The result matches the number of rows of the first matrix and columns of the second matrix.

Practicing matrix multiplication reinforces understanding and ensures you master this critical concept.