Problem 30 - Precalculus: Functions and Graphs

Question

If \(A=\left(a_{i j}\right)\) is any \(2 \times 2\) matrix such that \(|A| \neq 0,\) show that \(A\) has an inverse, and find a general formula for \(A^{-1}\).

Step-by-Step Solution

Answer

The inverse of a 2x2 matrix \(A\) is \(A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\).

Step 1: Definition of an Inverse

For any given matrix \(A\), an inverse \(A^{-1}\) exists if the product \(AA^{-1} = I\), where \(I\) is the identity matrix. For a \(2 \times 2\) matrix, the identity matrix is \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \). Given that \(|A| eq 0\) implies \(A\) is invertible.

Step 2: General Formula for the Determinant of a 2x2 Matrix

Consider a matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\). The determinant \(|A|\) is calculated as \(ad - bc\). Since \(|A| eq 0\), \(ad - bc eq 0\), thus \(A\) is non-singular and has an inverse.

Step 3: Finding the Inverse of a 2x2 Matrix

The inverse of matrix \(A\), denoted \(A^{-1}\), is given by the formula:\[ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \].This formula is derived by swapping \(a\) and \(d\), negating \(b\) and \(c\), then multiplying by \(\frac{1}{ad-bc}\).

Step 4: Verify the Inverse Formula

To confirm the formula, multiply \(A\) and \(A^{-1}\):\[ A A^{-1} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} d/(ad-bc) & -b/(ad-bc) \ -c/(ad-bc) & a/(ad-bc) \end{pmatrix} \].This product equals the identity matrix \(I\), thus verifying \(A A^{-1} = I\).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix of the form \[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]The determinant, denoted as \(|A|\), is calculated using the formula:\[|A| = ad - bc\]This value gives us important information about the matrix. If the determinant is not zero, the matrix is said to be "non-singular" and has an inverse. In contrast, if the determinant is zero, the matrix is "singular," meaning it does not have an inverse. In simple terms, the determinant tells you whether a matrix can be inverted or not.
Therefore, calculating the determinant is a crucial first step in finding out if a matrix is invertible.

Identity Matrix

An identity matrix is a special type of matrix that has ones on the main diagonal and zeros elsewhere. For a 2x2 matrix, the identity matrix is represented as:\[I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\]The identity matrix is important because when any matrix is multiplied by it, the original matrix is obtained. In essence, it behaves like the number "1" in matrix arithmetic. When dealing with matrix inversion, verifying that\[AA^{-1} = I\]vouches that \(A^{-1}\) is indeed the inverse of \(A\).
Understanding the role of the identity matrix helps clarify the concept of matrix inversion as a process of "undoing" a transformation represented by the original matrix.

Inverse of a 2x2 Matrix

Finding the inverse of a 2x2 matrix involves a well-defined process, provided the matrix is non-singular. For the matrix:\[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]The formula for the inverse, \(A^{-1}\), is given by:\[A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}\]Here’s how it works:

• First, calculate the determinant \(|A| = ad - bc\).
• Provided \(|A|\) is not zero, swap the positions of \(a\) and \(d\).
• Negate \(b\) and \(c\).
• Multiply the resulting matrix by \(1/|A|\).

By using this formula, you ensure that multiplying the original matrix \(A\) with its inverse \(A^{-1}\) yields the identity matrix. This is a practical and essential operation in various fields like computer graphics, physics, and economics.

Non-singular Matrix

A non-singular matrix, also known as an invertible matrix, is a matrix that has a non-zero determinant. For a 2x2 matrix:\[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]If the determinant \(ad - bc eq 0\), then the matrix is considered non-singular. This means it possesses an inverse, allowing us to "reverse" the matrix operation. In geometric terms, a non-singular matrix does not map any vectors to zero vectors, maintaining the dimensionality of the space. Therefore, non-singular matrices are crucial where matrix operations must be reversible, such as solving systems of equations or transforming spaces in various scientific applications.
Always start by confirming that a matrix is non-singular before attempting to compute its inverse.