Problem 34 - Differential Equations and Linear Algebra

Question

An invertible linear transformation \(\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is given. Find a formula for the inverse linear transformation. \(T_{3}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) defined by \(T_{3}(\mathbf{x})=A \mathbf{x},\) where $$A=\left[\begin{array}{lll} 3 & 5 & 1 \\ 1 & 2 & 1 \\ 2 & 6 & 7 \end{array}\right]$$

Step-by-Step Solution

Answer

To find the inverse linear transformation $T_{3}^{-1}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$, we first computed the inverse of the given matrix A as: \[A^{-1} = \begin{bmatrix} -\frac{4}{9} & \frac{29}{18} & -\frac{1}{6}\\ \frac{5}{18} & -\frac{10}{9} & -\frac{1}{18}\\ -\frac{1}{9} & \frac{4}{9} & \frac{2}{9} \end{bmatrix}\] Then, we define the inverse linear transformation $T_{3}^{-1}(\textbf{x})$ as: \(T_3^{-1}(\textbf{x}) = A^{-1}\textbf{x}\) with the above matrix as the inverse transformation matrix.

Step 1: Determine the inverse of matrix A

To determine the inverse of matrix A, use the following formula: \[A^{-1} = \frac{1}{det(A)} adj(A)\] where 'det(A)' is the determinant of A and 'adj(A)' is the adjugate of A. Let's compute the determinant of A first.

Step 2: Calculate the determinant of matrix A

To calculate the determinant, use the following formula for a 3x3 matrix: \[det(A) = \begin{vmatrix} 3 & 5 & 1\\ 1 & 2 & 1\\ 2 & 6 & 7 \end{vmatrix} = 3\begin{vmatrix} 2& 1\\ 6& 7 \end{vmatrix} - 5\begin{vmatrix} 1& 1 \\ 2& 7 \end{vmatrix} + 1\begin{vmatrix} 1& 2\\ 2& 6 \end{vmatrix}\] After completing the calculations, you should get: \[ det(A) = 3(2)(7-6) - 5(1)(7-2) + 1(1)(6-4) = 3(1) - 5(5) + 1(2) = -18 \]

Step 3: Calculate the adjugate of matrix A

To find the adjugate of a 3x3 matrix, follow these steps: 1. Calculate the matrix of co-factors of A. 2. Transpose the resulting matrix to obtain the adjugate. The matrix of co-factors (C) is given by: \[C = \begin{bmatrix} \begin{vmatrix} 2& 1\\ 6& 7 \end{vmatrix} & -\begin{vmatrix} 1& 1\\ 2& 7 \end{vmatrix}& \begin{vmatrix} 1& 2\\ 2& 6 \end{vmatrix}\\\\ -\begin{vmatrix} 5& 1\\ 6& 7 \end{vmatrix}& \begin{vmatrix} 3& 1\\ 2& 7 \end{vmatrix} & -\begin{vmatrix} 3& 5\\ 2& 6 \end{vmatrix}\\\\ \begin{vmatrix} 5& 1\\ 2& 1 \end{vmatrix} & -\begin{vmatrix} 3& 1\\ 1& 1 \end{vmatrix}& \begin{vmatrix} 3& 5\\ 1& 2 \end{vmatrix}\\ \end{bmatrix}\] After completing the calculations, we get: \[C = \begin{bmatrix} (14-6) & -(7-2) & (6-4)\\ -(35-6) & (21-1) & -(18-10)\\ 5-2 & -2+1 & 1-5 \end{bmatrix} = \begin{bmatrix} 8 & -5 & 2\\ -29 & 20 & -8\\ 3 & 1 & -4 \end{bmatrix}\] Now, transpose matrix C to get the adjugate of A: \[adj(A) = \begin{bmatrix} 8 & -29 & 3 \\ -5 & 20 & 1 \\ 2 & -8 & -4 \end{bmatrix}\]

Step 4: Calculate the inverse of matrix A

Now that we have the determinant and adjugate of A, we can calculate the inverse matrix A using the formula: \[A^{-1} = \frac{1}{-18} \begin{bmatrix} 8 & -29 & 3 \\ -5 & 20 & 1 \\ 2 & -8 & -4 \end{bmatrix}\] After multiplying by the scalar, we get: \[A^{-1} = \begin{bmatrix} -\frac{4}{9} & \frac{29}{18} & -\frac{1}{6}\\ \frac{5}{18} & -\frac{10}{9} & -\frac{1}{18}\\ -\frac{1}{9} & \frac{4}{9} & \frac{2}{9} \end{bmatrix}\]

Step 5: Define the inverse linear transformation

With the inverse matrix A found, we can define the inverse linear transformation T_3^{-1} as follows: \[T_3^{-1}(\textbf{x}) = A^{-1}\textbf{x}\] where \[A^{-1} = \begin{bmatrix} -\frac{4}{9} & \frac{29}{18} & -\frac{1}{6}\\ \frac{5}{18} & -\frac{10}{9} & -\frac{1}{18}\\ -\frac{1}{9} & \frac{4}{9} & \frac{2}{9} \end{bmatrix}\]

Key Concepts

Matrix Inverse

The concept of a matrix inverse is similar to finding the reciprocal of a number. If you have a matrix \(A\), its inverse \(A^{-1}\) is such that when multiplied together, they result in the identity matrix \(I\). In mathematical terms, this is expressed as \(AA^{-1} = A^{-1}A = I\). Determining the inverse of a matrix allows us to solve systems of linear equations and find transformations in reverse.

To find the inverse of a matrix, we use the formula \(A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)\). This formula tells us that the inverse depends on two key components: the determinant and the adjugate of the matrix.

Keep in mind:

• The matrix must be square, meaning it has the same number of rows and columns.
• The determinant of the matrix must not be zero; otherwise, the matrix does not have an inverse.

Determinant of a Matrix

To understand the inverse, you first need to calculate the determinant of a matrix, especially in the context of a 3x3 matrix. The determinant helps measure the volume distortion caused by the matrix when it acts as a transformation. For a 3x3 matrix \(A\), calculating the determinant involves finding the minors and applying a specific formula.

For our matrix \(A\), the determinant \(\det(A)\) is calculated by expanding along the first row:

• \(\det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\).

In these expressions, \(a, b, ... ,i\) are the elements of the matrix. If \(\det(A)\) equals zero, then \(A\) is singular, and it does not have an inverse.

Here’s why the determinant matters:

• Without a non-zero determinant, you cannot find the inverse.
• Understanding the determinant can provide insights into the properties of a matrix and its transformations.

Adjugate Matrix

The adjugate matrix, also known as the adjoint, is part of finding the inverse. It involves two main steps: finding the matrix of cofactors, and then transposing it.

To obtain the matrix of cofactors:

• Each entry is formed by calculating the determinant of the submatrix obtained by removing the row and column of that entry. The sign of the cofactor alternates according to the position.

After finding the cofactor matrix, you transpose it to get the adjugate. The transpose involves flipping the matrix over its diagonal, essentially switching its rows with columns.

This process is crucial because:

• The adjugate plays a direct role in the calculation of the inverse.
• It makes the matrix inversion process systematic and organized.

Understanding the adjugate matrix and its role bridges the gap between simple transformations and more complex inverse calculations.