Problem 26 - Algebra for College Students

Question

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rr} -3 & 1 \\ 3 & -2 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix is \( \begin{bmatrix} \frac{-2}{3} & \frac{-1}{3} \\ -1 & -1 \end{bmatrix} \).

Step 1: Calculate the Determinant

To find the inverse of a matrix, it must be invertible, which means its determinant should not be zero. The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). For the matrix \( \begin{bmatrix} -3 & 1 \ 3 & -2 \end{bmatrix} \), calculate the determinant as follows: \[-3(-2) - (1)(3) = 6 - 3 = 3\].

Step 2: Check Invertibility

Since the determinant we calculated in Step 1 is 3 (which is not zero), the matrix is invertible. Thus, we can proceed to find its inverse.

Step 3: Formulate the Inverse Matrix

The formula for finding the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Use this formula for our specific matrix: \[ \frac{1}{3} \begin{bmatrix} -2 & -1 \ -3 & -3 \end{bmatrix} \].

Step 4: Simplify the Inverse Matrix

Multiply each entry in the matrix by \( \frac{1}{3} \) to simplify:\[ \begin{bmatrix} \frac{-2}{3} & \frac{-1}{3} \ -1 & -1 \end{bmatrix} \]. This is the multiplicative inverse of the original matrix.

Key Concepts

Determinant of a matrix

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, the determinant helps us determine if a matrix is invertible. If the determinant is zero, the matrix cannot be inverted. For any 2x2 matrix, written as:

• \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]

We calculate its determinant using the formula:

• \[ad - bc\]

Here, in the example matrix\[\begin{bmatrix} -3 & 1 \ 3 & -2 \end{bmatrix}\],we compute:

• \[(-3 imes -2) - (1 imes 3) = 6 - 3 = 3\]

Since this result is not zero, the matrix may have an inverse.

2x2 Matrix

A 2x2 matrix is a simple form of a mathematical matrix. It has two rows and two columns, making it straightforward to work with, especially for newcomers to linear algebra. Often represented as:

• \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]

Every element within the matrix corresponds to a specific position and can significantly affect its properties when changed. Understanding how to manipulate these matrices through operations such as determinant calculation can solve various engineering and computing problems. In many cases, learning to compute the determinant and inverse of a 2x2 matrix sets the groundwork for understanding more complex operations in higher-dimensional matrices.

Matrix Invertibility

Matrix invertibility is the condition that tells us whether a matrix has an inverse. In simpler terms, it's the ability to "reverse" a matrix operation. For a matrix to be invertible, its determinant must be a non-zero value. When dealing specifically with 2x2 matrices, determining invertibility involves:

• Calculating the determinant using the formula: \[ad - bc\]
• Ensuring the result is non-zero.

Suppose we compute the determinant of our example matrix and find that it's 3. In that case, because this number is not zero, we proceed to find the inverse: \[\frac{1}{3}\begin{bmatrix} -2 & -1 \ -3 & -3 \end{bmatrix}\].Finally, simplify each element by multiplying by \(\frac{1}{3}\). The existence of an inverse allows us to solve equations where this matrix is involved, effectively "undoing" its effects. This inverse can be a powerful tool in solving systems of linear equations.