Problem 60 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Check if the Matrix is Invertible

For a 2x2 matrix, \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the matrix is invertible if the determinant \( \det(A) eq 0 \). Calculate the determinant: \( \det(A) = ad - bc \). For matrix \( A \), \( a = 1 \), \( b = 2 \), \( c = 1 \), \( d = 3 \). So, \( \det(A) = 1(3) - 1(2) = 3 - 2 = 1 \). Since the determinant is not zero, the matrix is invertible.

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

The inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Substitute \( a, b, c, \) and \( d \) from \( A \) into this formula: \( A^{-1} = \frac{1}{1} \begin{bmatrix} 3 & -2 \ -1 & 1 \end{bmatrix} \).

Step 3: Simplify the Inverse Matrix

Since the determinant \( \det(A) = 1 \), it does not affect the entries of the inverse matrix during division. Therefore, the inverse matrix is simply \( A^{-1} = \begin{bmatrix} 3 & -2 \ -1 & 1 \end{bmatrix} \).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can tell us a lot about the matrix. For a simple 2x2 matrix, the determinant is calculated using the formula \( \det(A) = ad - bc \), where \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \). The determinant helps determine if a matrix is invertible. If the determinant is zero, the matrix is not invertible. This is because a zero determinant means the matrix does not have an inverse. In practice, computing the determinant of a 2x2 matrix is straightforward: you multiply the entries on the main diagonal and subtract the product of the other two entries. This small calculation gives great insight into the properties of the matrix.

2x2 Matrix

A 2x2 matrix is one of the simplest types of matrices you’ll encounter in linear algebra. It consists of two rows and two columns. Mathematically, it's represented in the form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Each element within the matrix has a specific place:

• \( a \): located at the first row, first column
• \( b \): located at the first row, second column
• \( c \): located at the second row, first column
• \( d \): located at the second row, second column

The simplicity of a 2x2 matrix makes it user-friendly for understanding the basic properties of matrices, such as their determinant and inverse. These matrices can be used to model simple systems or transformations in two dimensions.

Invertible Matrix

An invertible matrix, sometimes called a non-singular or non-degenerate matrix, is a matrix that has an inverse. For a 2x2 matrix, determining if it is invertible is a matter of checking that the determinant is not zero. When a matrix is invertible, you can find another matrix that, when multiplied with the original, yields the identity matrix (a matrix with 1s on its main diagonal and 0s elsewhere). Finding the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) involves using the formula:
\[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]
This rearranges the positions and signs of the elements in the matrix to create the inverse. The inverse allows solving linear equations where the matrix represents the system's coefficients by enabling us to work back to original values. It plays a crucial role in various applications such as statistics, physics, and computer graphics.