Problem 62 - Calculus for Biology and Medicine

Question

Write down the inverse of \(A\). $$ A=\left[\begin{array}{rr} -2 & -1 \\ 3 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

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

Step 1: Determine Adjoint

For a 2x2 matrix, the adjoint is found by swapping the diagonal elements and changing the signs of the off-diagonal elements. Starting with matrix:\[A = \begin{bmatrix} -2 & -1 \ 3 & 2 \end{bmatrix}\]The adjoint (or adjugate), \( ext{adj}(A) \), will be:\[\text{adj}(A) = \begin{bmatrix} 2 & 1 \ -3 & -2 \end{bmatrix}\]

Step 2: Compute the Determinant

The determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using:\[\text{det}(A) = ad - bc\]Substitute the values from matrix \( A \):\[\text{det}(A) = (-2)(2) - (-1)(3) = -4 + 3 = -1\]

Step 3: Find Inverse Using Determinant and Adjoint

The formula for the inverse of matrix \( A \) is given by:\[A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)\]Substitute the determined values:\[A^{-1} = \frac{1}{-1} \cdot \begin{bmatrix} 2 & 1 \ -3 & -2 \end{bmatrix} = \begin{bmatrix} -2 & -1 \ 3 & 2 \end{bmatrix}\]

Step 4: Solution Verification

Verify that the multiplication of \( A \) and \( A^{-1} \) produces the identity matrix \( I_2 \), confirming the inverse is correct:\[A \cdot A^{-1} = \begin{bmatrix} -2 & -1 \ 3 & 2 \end{bmatrix} \cdot \begin{bmatrix} -2 & -1 \ 3 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]This shows \( A^{-1} \) is indeed correct.

Key Concepts

Adjoint Matrix

The adjoint matrix, also known as the adjugate matrix, is a crucial component when finding the inverse of a given matrix. Especially in the case of a 2x2 matrix, the process to determine the adjoint is straightforward. For a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the adjoint is determined by swapping the positions of the elements on the main diagonal and changing the signs of the off-diagonal elements. Specifically,

• Swap the diagonal elements: \( a \) and \( d \).
• Change the sign of the off-diagonal elements: \( b \) and \( c \).

This results in the adjoint matrix: \[ \text{adj}(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] This matrix plays a key role in calculating the inverse, as it is one of the two primary components used in the inverse formula together with the determinant.

Determinant

The determinant of a matrix provides a scalar value that signifies several important properties of the matrix, such as whether the matrix has an inverse. For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula: \[ \text{det}(A) = ad - bc \] The determinant must be non-zero for the matrix to have an inverse. When it equals zero, the matrix is said to be singular and thus, doesn't have an inverse. In our example, the determinant was calculated as follows: \[ \text{det}(A) = (-2)(2) - (-1)(3) = -4 + 3 = -1 \] Since the determinant \( \text{det}(A) \) is not zero, we can proceed to find the inverse of the matrix using this value in the formula: \[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \] The determinant thus serves as a scaling factor in the calculation of the matrix inverse.

Identity Matrix

The identity matrix, often denoted as \( I \), plays a pivotal role in linear algebra. It is an essential concept in understanding matrix operations and the verification of a matrix inverse. For any square matrix, there exists an identity matrix of the same dimensions, which acts similarly to the number 1 in regular multiplication, meaning: multiplying any matrix by the identity matrix results in the original matrix.For a 2x2 matrix, the identity matrix \( I_2 \) looks like this: \[ I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \] When verifying the correctness of a calculated matrix inverse, we multiply the original matrix by its presumed inverse. If the product is the identity matrix, \( A \cdot A^{-1} = I \), this confirms that \( A^{-1} \) is indeed the correct inverse. In practice, this means every element in the resulting product matrix matches the elements in \( I_2 \). It serves as a reliable check-point for matrix inversion.