Problem 61 - Calculus for Biology and Medicine

Question

Use the determinant to find the inverse of \(A\) $$ A=\left[\begin{array}{rr} -1 & 4 \\ 5 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \( \begin{bmatrix} -\frac{1}{21} & \frac{4}{21} \\ \frac{5}{21} & \frac{1}{21} \end{bmatrix} \).

Step 1: Calculate the Determinant of Matrix A

The determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula \( \text{det}(A) = ad - bc \). For our matrix \( A = \begin{bmatrix} -1 & 4 \ 5 & 1 \end{bmatrix} \), we have \( a = -1 \), \( b = 4 \), \( c = 5 \), \( d = 1 \). Substituting these values into the formula, we get \( \text{det}(A) = (-1)(1) - (4)(5) = -1 - 20 = -21 \).

Step 2: Check if the Matrix is Invertible

A square matrix is invertible if and only if its determinant is non-zero. Since the determinant of matrix \( A \) is \( -21 \), which is non-zero, the matrix is invertible.

Step 3: Find the Adjugate of Matrix A

The adjugate of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( \text{adj}(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Substituting the values from matrix \( A = \begin{bmatrix} -1 & 4 \ 5 & 1 \end{bmatrix} \), we find \( \text{adj}(A) = \begin{bmatrix} 1 & -4 \ -5 & -1 \end{bmatrix} \).

Step 4: Calculate the Inverse of Matrix A

The inverse of a matrix \( A \) is given by \( A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A) \). From Step 1, we found that \( \text{det}(A) = -21 \) and from Step 3, \( \text{adj}(A) = \begin{bmatrix} 1 & -4 \ -5 & -1 \end{bmatrix} \). Therefore, \( A^{-1} = \frac{1}{-21} \times \begin{bmatrix} 1 & -4 \ -5 & -1 \end{bmatrix} = \begin{bmatrix} \frac{1}{-21} & \frac{-4}{-21} \ \frac{-5}{-21} & \frac{-1}{-21} \end{bmatrix} \). Simplifying, \( A^{-1} = \begin{bmatrix} -\frac{1}{21} & \frac{4}{21} \ \frac{5}{21} & \frac{1}{21} \end{bmatrix} \).

Key Concepts

Determinant Calculation

The determinant of a matrix plays a crucial role in various aspects of linear algebra, including determining whether a matrix is invertible. For a 2x2 matrix like \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix}, \]the determinant is calculated using the formula:\[ \text{det}(A) = ad - bc. \]In the example matrix \(A = \begin{bmatrix} -1 & 4 \ 5 & 1 \end{bmatrix},\)substitute the values where \(a = -1\), \(b = 4\), \(c = 5\), and \(d = 1\).
This gives us:\[ \text{det}(A) = (-1)(1) - (4)(5) = -1 - 20 = -21. \]The determinant not only helps find the inverse but also checks if a matrix is invertible. A non-zero determinant indicates the matrix can indeed be inverted, as shown here.

Adjugate Matrix

The adjugate of a 2x2 matrix is used when finding the matrix inverse, especially in manual calculations. It's derived by swapping the diagonal elements and changing the signs of the off-diagonal elements. For a general 2x2 matrix:\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix}, \]the adjugate is:\[ \text{adj}(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix}. \]Applying this to our specific example matrix \(A = \begin{bmatrix} -1 & 4 \ 5 & 1 \end{bmatrix}\),we find the adjugate matrix:\[ \text{adj}(A) = \begin{bmatrix} 1 & -4 \ -5 & -1 \end{bmatrix}. \]Using the adjugate is a key step in ultimately determining the inverse of a matrix.

Invertible Matrix Criteria

Understanding when a matrix is invertible is fundamental in linear algebra. A square matrix is said to be invertible—or non-singular—if its determinant is not equal to zero. This criterion ensures that a unique inverse matrix exists. The inverse is utilized in solving linear equations and transforming systems.
For our example matrix \(A = \begin{bmatrix} -1 & 4 \ 5 & 1 \end{bmatrix}\),we calculated that \(\text{det}(A) = -21\),which is non-zero. This confirms that matrix \(A\) is indeed invertible.
An invertible matrix allows for meaningful computations such as finding solutions to linear systems, which is often represented as\[ A \mathbf{x} = \mathbf{b}. \]Knowing your matrix is invertible gives the green light for calculating other important characteristics in linear transformations.