Problem 33 - 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. \(T_{2}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by \(T_{2}(\mathbf{x})=A \mathbf{x},\) where $$A=\left[\begin{array}{ll} 1 & 1 \\ 2 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of the given matrix \(A=\begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}\) is \(A^{-1} = \begin{bmatrix} -3 & 1 \\ 2 & -1 \end{bmatrix}\). Therefore, the inverse linear transformation \(T_{2}^{-1}\) is given by the formula \(T_{2}^{-1}(\mathbf{x}) = \begin{bmatrix} -3 & 1 \\ 2 & -1 \end{bmatrix} \mathbf{x}\).

Step 1: Verify if the Matrix is Invertible

The given matrix \(A\) is invertible if the determinant of the matrix is not equal to zero. The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by \(ad - bc\). So, calculate the determinant of the given matrix \(A\): Det\((A) = 1 \cdot 3 - 1 \cdot 2 = 1\) Since the determinant of the matrix \(A\) is not equal to zero, the matrix \(A\) is invertible.

Step 2: Calculate the Inverse of the Matrix

The formula for the inverse of a 2x2 matrix \(A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is \(A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\). Substitute \(a = 1, b = 1, c = 2, d = 3\) into the formula: \(A^{-1} = \frac{1}{1-2} \begin{bmatrix} 3 & -1 \\ -2 & 1 \end{bmatrix}\) You will obtain: \(A^{-1} = \begin{bmatrix} -3 & 1 \\ 2 & -1 \end{bmatrix}\)

Step 3: Write the Formula for the Inverse Transformation

Finally, write the formula for the inverse transformation \(T_{2}^{-1}\). The inverse transformation is given by \(T_{2}^{-1}(\mathbf{x}) = A^{-1} \mathbf{x}\). With \(A^{-1} = \begin{bmatrix} -3 & 1 \\ 2 & -1 \end{bmatrix}\), you get: \(T_{2}^{-1}(\mathbf{x}) = \begin{bmatrix} -3 & 1 \\ 2 & -1 \end{bmatrix} \mathbf{x}\) Which is the inverse of the given linear transformation.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special value that can tell us a lot about the matrix itself. For a 2x2 matrix, it's quite straightforward to calculate. The determinant helps us understand whether a matrix is invertible, among other things.
For a general 2x2 matrix of the form:

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

We use the formula:

• Determinant \( = ad - bc \)

This formula takes the product of the diagonal elements \( a \) and \( d \), and subtracts the product of the off-diagonal elements \( b \) and \( c \).
In our problem, the matrix \( A \) is:

• \( A = \begin{bmatrix} 1 & 1 \ 2 & 3 \end{bmatrix} \)

So, the calculation is:

• \( Det(A) = 1 \cdot 3 - 1 \cdot 2 = 1 \)

Since the determinant is not zero, this matrix has an inverse. A non-zero determinant implies that the matrix transformation scales areas and is invertible.

Matrix Inverse

Finding the inverse of a matrix is like finding its reverse transformation—it allows us to go backwards. When we multiply a matrix by its inverse, we get the identity matrix, which behaves like multiplying by one.For a 2x2 matrix \( A \) with elements \( a, b, c, \) and \( d \):

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

The inverse \( A^{-1} \) is given by:

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

The term \( ad-bc \) is the determinant, and it must not be zero for the inverse to exist.
For the matrix \( A = \begin{bmatrix} 1 & 1 \ 2 & 3 \end{bmatrix} \), we substitute to find:

• \( A^{-1} = \frac{1}{1} \begin{bmatrix} 3 & -1 \ -2 & 1 \end{bmatrix} = \begin{bmatrix} 3 & -1 \ -2 & 1 \end{bmatrix} \)

Having the inverse, you can determine the action needed to reverse the transformation applied by \( A \). This is crucial in applications where reversing a process or transformation is needed.

2x2 Matrix

The 2x2 matrix is a simple yet powerful tool used in linear algebra to represent linear transformations in a two-dimensional space. Unlike higher-dimensional matrices, 2x2 matrices are compact and are easy to visualize and understand. They are often used in processes like transformations, rotations, and scaling.A matrix like:

• \( \begin{bmatrix} 1 & 1 \ 2 & 3 \end{bmatrix} \)

Represents a linear mapping from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \).When applied to a vector \( \mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix} \), the result is a new vector obtained by:

• Performing linear combinations of its components.

This makes the 2x2 matrix a cornerstone of two-dimensional linear transformations like scaling, rotating, or skewing objects. Understanding the role and functions of the 2x2 matrix is fundamental in fields such as computer graphics, physics, and engineering where space transformations are commonplace.