Problem 43 - Calculus for Biology and Medicine

Question

Show that the inverse of $$A=\left[\begin{array}{ll} 2 & 1 \\ 1 & 3 \end{array}\right]$$ is $$B=\frac{1}{5}\left[\begin{array}{rr} 3 & -1 \\ -1 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of \(A\) is \(B\).

Step 1: Write down the formula for the inverse of a 2x2 matrix

For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the inverse \(A^{-1}\) is given by \( \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \), provided \(ad - bc eq 0\).

Step 2: Compute the determinant of A

Compute the determinant of the matrix \(A = \begin{bmatrix} 2 & 1 \ 1 & 3 \end{bmatrix}\) using the formula \(ad - bc\). Here, \(a = 2\), \(b = 1\), \(c = 1\), and \(d = 3\). So, \(det(A) = 2 \cdot 3 - 1 \cdot 1 = 6 - 1 = 5\).

Step 3: Apply the inverse formula using the determinant

Substitute \(a = 2\), \(b = 1\), \(c = 1\), and \(d = 3\) into the inverse formula to get \(A^{-1} = \frac{1}{5} \begin{bmatrix} 3 & -1 \ -1 & 2 \end{bmatrix}\).

Step 4: Compare with given matrix B

The given matrix \(B\) is \(\frac{1}{5} \begin{bmatrix} 3 & -1 \ -1 & 2 \end{bmatrix}\), which matches our computed inverse. Thus, the inverse \(A^{-1} = B\) is confirmed.

Key Concepts

Understanding the Determinant

Calculating the determinant is a crucial step in finding the inverse of a matrix, particularly for a 2x2 matrix.The determinant of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is calculated using the formula \(ad - bc\).This value gives insight into the characteristics of the matrix, such as whether it is invertible or not.

• A matrix is invertible only if its determinant is non-zero.

For our specific matrix \(A = \begin{bmatrix} 2 & 1 \ 1 & 3 \end{bmatrix}\), the determinant is calculated as \(2 \times 3 - 1 \times 1 = 6 - 1 = 5\).Since the determinant is 5, which is not zero, we can find its inverse.

Spotlight on the 2x2 Matrix

A 2x2 matrix is a matrix that consists of two rows and two columns.It can represent a variety of transformations when used in linear algebra.

• Each element in the matrix plays a role in calculations like matrix multiplication and finding the determinant.

The structure of a 2x2 matrix makes it simple to manipulate by following standard formulas.For the matrix \(A = \begin{bmatrix} 2 & 1 \ 1 & 3 \end{bmatrix}\), the numbers in the top-left to bottom-right diagonal (\(2\) and \(3\)) are known as diagonal elements.Understanding these elements is key, especially when calculating the determinant and inverse.

Demystifying Matrix Multiplication

Matrix multiplication is an operation that allows combining two matrices.For the inverse finding process, checking that the product of a matrix and its supposed inverse results in the identity matrix is essential.

• The identity matrix for a 2x2 is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).
• To test if \(B = \frac{1}{5}\begin{bmatrix} 3 & -1 \ -1 & 2 \end{bmatrix}\) is the inverse of \(A\), we perform matrix multiplication.

By multiplying \(A\) and \(B\), the result should be the identity matrix:\[\begin{bmatrix} 2 & 1 \ 1 & 3 \end{bmatrix} \times \frac{1}{5}\begin{bmatrix} 3 & -1 \ -1 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]If this condition holds true, matrix \(B\) is indeed the inverse of matrix \(A\). These steps show the power and necessity of matrix multiplication when validating inverses in linear algebra.