Problem 8 - Algebra for College Students

Question

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rr} 5 & -1 \\ 3 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse matrix is \(\left[\begin{array}{cc}\frac{4}{23} & \frac{1}{23} \\ \frac{-3}{23} & \frac{5}{23}\end{array}\right]\).

Step 1: Determine if the Matrix is Invertible

To find the multiplicative inverse of a matrix, we first need to determine if the given matrix is invertible. A matrix is invertible if its determinant is not zero. For the matrix \(\left[\begin{array}{rr}5 & -1 \ 3 & 4\end{array}\right]\), the determinant is calculated as: \(det(A) = 5 \cdot 4 - (-1) \cdot 3 = 20 + 3 = 23\). Since the determinant is 23, the matrix is invertible.

Step 2: Formulate the Inverse Formula

The inverse of a 2x2 matrix \(\left[\begin{array}{cc}a & b \ c & d\end{array}\right]\) is given by \(\frac{1}{ad - bc}\left[\begin{array}{cc}d & -b \ -c & a\end{array}\right]\), provided the determinant \(ad - bc\) is not zero.

Step 3: Apply the Inverse Formula

Now that we know the determinant of our matrix is 23, we can apply the inverse formula. Substitute the values from the matrix \(\left[\begin{array}{rr}5 & -1 \ 3 & 4\end{array}\right]\) into the formula: \[A^{-1} = \frac{1}{23}\left[\begin{array}{cc}4 & 1 \ -3 & 5\end{array}\right]\].

Step 4: Write the Inverse Matrix

Finally, dividing each element in the matrix \(\left[\begin{array}{cc}4 & 1 \ -3 & 5\end{array}\right]\) by 23, we get the inverse matrix: \[A^{-1} = \left[\begin{array}{cc}\frac{4}{23} & \frac{1}{23} \ \frac{-3}{23} & \frac{5}{23}\end{array}\right]\].

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that we calculate from its elements. It's essential in determining if a matrix possesses an inverse. For a 2x2 matrix, the determinant is calculated using the formula:

• For a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is \( ad - bc \).

In the given example, the matrix\[\left[\begin{array}{rr}5 & -1 \ 3 & 4\end{array}\right]\]has a determinant evaluated as:

• \( 5 \cdot 4 - (-1) \cdot 3 = 20 + 3 = 23 \).

Because the determinant is 23, which is not zero, the matrix qualifies as invertible. This makes the calculation of both determinants and their presence in matrix theory crucial for further exploration of matrix properties.

Invertible Matrix

An invertible matrix, also known as a non-singular or nonsingular matrix, is one that has an inverse. This means there is another matrix that, when multiplied by the original, gives the identity matrix. The identity matrix is similar to the number 1 in multiplication but in matrix form. An important criteria for a matrix to be invertible is that its determinant must not be zero.
Here’s why the determinant is important:

• If the determinant is zero, it implies that the matrix is singular and does not have an inverse.
• Non-zero determinant indicates the matrix is invertible.

In this exercise, the determinant of the given matrix was 23, confirming it is invertible, allowing us to proceed with finding its inverse. Understanding the conditions under which a matrix is invertible is foundational for solving systems of linear equations and more advanced calculations.

2x2 Matrix

A 2x2 matrix consists of two rows and two columns. Their simplicity makes them a common introduction point for matrix concepts, including determinant calculation and finding inverses. The general form of a 2x2 matrix is represented as:

• \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\)

The process to find the inverse of such a matrix involves the formula:

• For the matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), its inverse is \(\frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\), assuming \( ad-bc eq 0 \).

In this exercise, we used these properties to successfully find the inverse of\(\begin{bmatrix} 5 & -1 \ 3 & 4 \end{bmatrix}\).By substituting into the inverse formula, we deduced that:

• \( A^{-1} = \frac{1}{23}\begin{bmatrix} 4 & 1 \ -3 & 5 \end{bmatrix} \)

This calculation highlights the approachable nature of 2x2 matrices, making them ideal for practices involving inverses and other matrix operations.