Problem 43 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

Matrix multiplication verifies that \( A \times B = I \), so \( B \) is the inverse of \( A \).

Step 1: Understanding the Problem

We are given a matrix \( A \) and a matrix \( B \), and we need to show that \( B \) is the inverse of \( A \). This means we need to demonstrate that \( A \times B = I \), where \( I \) is the identity matrix for 2x2 matrices, which is \( \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right] \).

Step 2: Performing the Matrix Multiplication

Multiply the matrices \( A \) and \( B \):\[\begin{array}{c}A = \left[\begin{array}{cc} 2 & 1 \ 1 & 1 \end{array}\right], \B = \left[\begin{array}{rr} 1 & -1 \ -1 & 2 \end{array}\right]\end{array}\]To find the product \( A \times B \), calculate each element of the resulting matrix:- First row, first column: \( (2 \times 1) + (1 \times -1) = 2 - 1 = 1 \)- First row, second column: \( (2 \times -1) + (1 \times 2) = -2 + 2 = 0 \)- Second row, first column: \( (1 \times 1) + (1 \times -1) = 1 - 1 = 0 \)- Second row, second column: \( (1 \times -1) + (1 \times 2) = -1 + 2 = 1 \)

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, where two matrices combine to form a new matrix. It is crucial to understand the process to perform tasks such as finding an inverse.
When multiplying matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. This allows each element in the resulting matrix to be derived from the sum of products of corresponding elements from the rows of the first matrix and the columns of the second.

• To multiply two matrices, say matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and matrix \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\), you perform the following calculations for each element in the product matrix \(C = AB\):
• First row, first column: \( C_{11} = ae + bg \)
• First row, second column: \( C_{12} = af + bh \)
• Second row, first column: \( C_{21} = ce + dg \)
• Second row, second column: \( C_{22} = cf + dh \)

This method ensures that the new matrix accurately represents the transformation resulting from the multiplication of the original matrices.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a square matrix, having two rows and two columns. Despite its simplicity, a 2x2 matrix plays a significant role in various mathematical applications including transformations, solving linear equations, and physics.
A typical 2x2 matrix is represented as:

• \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

The determinant of a 2x2 matrix is calculated as \( ad - bc \). This value is essential for finding the inverse of the matrix, which is possible when the determinant is not zero.
To find the inverse of a 2x2 matrix, you use the formula:

• If \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is \( A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \)

This operation rearranges the elements of \( A \) using the determinant, allowing for the reverse of the transformation implied by the matrix.

Identity Matrix

The identity matrix is a special kind of matrix pivotal in matrix algebra, functioning similarly to the number 1 in multiplication for real numbers. In a 2x2 identity matrix, you have two rows and two columns, where the diagonal elements are 1, and all off-diagonal elements are 0.
A 2x2 identity matrix is written as:

• \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)

When any matrix is multiplied by the identity matrix, the original matrix is the result. This demonstrates the purpose of the identity matrix in maintaining the integrity of the values when matrix operations occur.
Whether it appears in a series of multiplicative operations or is used to verify inverses, the identity matrix serves as a benchmark. For a matrix \( A \), if \( A \times B = I \), then \( B \) is the inverse of \( A \). This is fundamental in solving systems of linear equations and other applications involving matrices.