Problem 44 - Calculus for Biology and Medicine

Question

Show that the inverse of $$A=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{array}\right]$$ is $$A^{-1}=\left[\begin{array}{ccc} \frac{3}{4} & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{2} & 1 & \frac{1}{2} \\ \frac{1}{4} & \frac{1}{2} & \frac{3}{4} \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of \( A \) is verified as given, \( A^{-1} = \begin{bmatrix} 3/4 & 1/2 & 1/4 \\ 1/2 & 1 & 1/2 \\ 1/4 & 1/2 & 3/4 \end{bmatrix}. \)

Step 1: Confirm Matrix Dimensions

First, verify that the matrix \( A \) is a square matrix. The matrix \( A \) is a 3x3 matrix, which means it is square, and therefore, it can potentially have an inverse.

Step 2: Find the Determinant of A

Calculate the determinant of matrix \( A \). For a 3x3 matrix \[ A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}, \]the determinant is \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg). \]So for our matrix, \[ \text{det}(A) = 2(4 - 1) - (-1)(-2) + 0 = 2 \times 3 - 1 = 5. \] Since the determinant is non-zero, the inverse exists.

Step 3: Use Adjugate Method to Find Inverse

Calculate the inverse using the formula: \[ A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A), \]where "adj(A)" is the adjugate of \( A \). The formula relates the inverse to the determinant and the adjugate matrix.

Step 4: Calculate Cofactor Matrix

Find the cofactor matrix of \( A \). Each element of this matrix is calculated by taking the determinant of the 2x2 submatrices that result when you remove the row and column of the element in question, then multiply by \((-1)^{i+j}\) where \(i\) and \(j\) are the element's row and column indices.

Step 5: Form the Adjugate Matrix

Transpose the cofactor matrix to form the adjugate matrix, \( \text{adj}(A) \).

Step 6: Calculate the Inverse Matrix

Divide each element of the adjugate matrix by the determinant (5 in this case) to get the inverse of \( A \):\[ A^{-1} = \frac{1}{5} \cdot \begin{bmatrix} 15 & 10 & 5 \ 10 & 20 & 10 \ 5 & 10 & 15 \end{bmatrix} = \begin{bmatrix} 3/4 & 1/2 & 1/4 \ 1/2 & 1 & 1/2 \ 1/4 & 1/2 & 3/4 \end{bmatrix}. \]

Step 7: Verify the Result

Multiply matrix \( A \) with its calculated inverse \( A^{-1} \). If the product is the identity matrix, the inverse is correct:\[ AA^{-1} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}. \]

Key Concepts

3x3 matrix

A **3x3 matrix** is a two-dimensional array of numbers arranged in three rows and three columns. Each element of this matrix can be referred to based on its position using row and column numbers. For example, in a matrix labeled as \( A \), the entry in the first row and second column is noted as \( a_{12} \). This structure is crucial because it ensures the matrix is square, a requirement for certain operations like finding an inverse.
The matrix you are dealing with here is:

• Row 1: \( [2, -1, 0] \)
• Row 2: \( [-1, 2, -1] \)
• Row 3: \( [0, -1, 2] \)

For a 3x3 matrix to have an inverse, it must be square, meaning that it has equal numbers of rows and columns, and its determinant should not be zero. If all these conditions are met, you can proceed with calculations like finding the matrix inverse. This matrix format is often used in systems of equations, transformations, and many more applications in mathematics and engineering.

determinant calculation

The **determinant** of a 3x3 matrix is a unique number that can tell us a lot about the matrix, including whether an inverse exists. If the determinant is 0, the matrix does not have an inverse. The formula for calculating the determinant of a 3x3 matrix \( A \) is: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]This formula uses elements from the matrix, arranged as follows:

• \( a, b, c \)
• \( d, e, f \)
• \( g, h, i \)

Taking the provided matrix example:

• \( A = \begin{bmatrix} 2 & -1 & 0 \ -1 & 2 & -1 \ 0 & -1 & 2 \end{bmatrix} \)
• \[ \text{det}(A) = 2(4 - 1) - (-1)(-2+0) \]
• \[ = 2 \times 3 - 1 = 5 \]

A determinant value of 5 confirms that the matrix is invertible, which means we can continue to find the inverse matrix using more advanced methods.

adjugate method

The **adjugate method** is a technique used to find the inverse of a matrix. Once you have the determinant of the matrix, the adjugate method involves calculating the cofactor matrix, transposing it, and then dividing by the determinant. It's an organized series of steps starting from the calculation of cofactors. Here's the process:

• First, create the cofactor matrix, where each element is the determinant of the 2x2 matrices after removing one row and one column.
• Next, transpose the cofactor matrix to get the adjugate matrix.
• Finally, the inverse matrix is calculated by taking \( \frac{1}{\text{det}(A)} \) of the adjugate matrix.

This method gives a structured approach to break down the inversion process into manageable chunks. This is particularly helpful when dealing with larger matrices like the 3x3 matrix in the exercise.

cofactor matrix

In the pursuit of finding a matrix's inverse using the adjugate method, the **cofactor matrix** plays a key role. Each element of this matrix is derived from a minor, which is the determinant of a smaller, usually 2x2, matrix formed by eliminating the row and column of the specific element from the original matrix. The steps to create a cofactor matrix include:

• Identify the minor for each element.
• Calculate the determinant of these minors.
• Apply a positive or negative sign based on the position using the formula \((-1)^{i+j}\), where \(i\) and \(j\) correspond to the row and column indices.

Once the cofactor matrix is computed, you take its transpose to get the adjugate matrix, necessary for finding the matrix inverse. Assembling a cofactor matrix is a crucial part of the matrix inversion process, dealing effectively with the complexities of removing specific rows and columns and calculating the needed determinants.