Problem 46 - Calculus for Biology and Medicine

Question

$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Find the inverse (if it exists) of \(B\).

Step-by-Step Solution

Answer

The inverse of \( B \) exists and is \( \left[ \begin{array}{cc} 0.2 & 0.2 \\ -0.3 & 0.2 \end{array} \right] \).

Step 1: Check if the Inverse Exists

To determine if the matrix \( B \) has an inverse, we need to calculate its determinant. The inverse of a matrix exists if and only if its determinant is non-zero. The determinant of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is given by the formula \( ad - bc \). For matrix \( B = \left[ \begin{array}{cc} 2 & -2 \ 3 & 2 \end{array} \right] \), we compute the determinant as follows:\[ \text{det}(B) = (2)(2) - (-2)(3) = 4 + 6 = 10 \]Since the determinant of \( B \) is \( 10 \), which is not zero, the inverse of \( B \) exists.

Step 2: Compute the Inverse of the Matrix

The formula for the inverse of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is \( \frac{1}{ad-bc} \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right] \). Using the values from matrix \( B = \left[ \begin{array}{cc} 2 & -2 \ 3 & 2 \end{array} \right] \), we substitute them into the formula:\[ B^{-1} = \frac{1}{10} \left[ \begin{array}{cc} 2 & 2 \ -3 & 2 \end{array} \right] \]The inverse of \( B \) is thus \( \left[ \begin{array}{cc} 0.2 & 0.2 \ -0.3 & 0.2 \end{array} \right] \), where each element is the result of dividing the corresponding element of the matrix by the determinant 10.

Key Concepts

Determinant Calculation

The determinant is a crucial concept in matrix algebra. It is especially important when determining whether a matrix has an inverse. For a 2x2 matrix, the determinant helps determine if the matrix is invertible.
The determinant of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is calculated using the formula:

• \( ad - bc \)

In our exercise, the matrix \( B \) is given as \( \left[ \begin{array}{cc} 2 & -2 \ 3 & 2 \end{array} \right] \). Calculating its determinant involves substituting the elements into the formula:

• \( 2 \cdot 2 - (-2) \cdot 3 = 4 + 6 = 10 \)

The determinant of \( B \) is 10, which is non-zero. This means that \( B \) is invertible, or in other words, it has an inverse.

2x2 Matrix Inverse

Finding the inverse of a 2x2 matrix is a straightforward process, once you've confirmed the determinant is non-zero. The formula for the inverse of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is:

• \( \frac{1}{ad-bc} \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right] \)

For our matrix \( B \), with a determinant of 10, the inverse \( B^{-1} \) is calculated by substituting values into this formula:

• \( \frac{1}{10} \left[ \begin{array}{cc} 2 & 2 \ -3 & 2 \end{array} \right] \)

The resulting inverse matrix is \( \left[ \begin{array}{cc} 0.2 & 0.2 \ -0.3 & 0.2 \end{array} \right] \). This matrix, when multiplied by the original, yields the identity matrix, confirming that it's the correct inverse.

Matrix Algebra

Matrix algebra provides powerful tools for working with systems of equations, among other applications. An important aspect of matrix algebra is the concepts of inverses and determinants, which are critical for operations such as solving linear systems.
In the world of matrices, the inverse of a matrix \( A \), symbolized as \( A^{-1} \), satisfies the equation \( A \cdot A^{-1} = I \), where \( I \) is the identity matrix. This is similar to how dividing by a number in arithmetic gives you the multiplicative identity 1.
The identity matrix acts as the multiplicative identity in matrix algebra. For a 2x2 matrix, it's represented as:

• \( \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \)

Having an inverse allows us to 'undo' matrix operations, similar to taking the arithmetic reciprocal. This is useful in various applications, from transforming coordinates to solving equations, making matrix algebra an indispensable part of many fields, including physics, computer science, and economics.