Problem 30 - Algebra and Trigonometry with Analytic Geometry

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

Matrix \( A \) has an inverse if \(|A| \neq 0\); the inverse is \( A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \).

Step 1: Understanding the Matrix and Determinant

We are given a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \). The determinant of matrix \( A \), denoted as \(|A|\), is calculated as \( ad - bc \). For the matrix to have an inverse, it is necessary that \(|A| eq 0\). This ensures that the matrix is non-singular and invertible.

Step 2: State the Formula for the Inverse

The inverse of a 2x2 matrix \( A\) is given by the formula:\[ A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]where \(|A| = ad - bc\), as long as \(|A| eq 0\). This formula shows the structure of the inverse matrix based on the elements of \( A \).

Step 3: Verify the Inverse Matches the Identity Matrix

To verify, compute \( A \cdot A^{-1} = I \) where \( I \) is the identity matrix:\[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \cdot \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} = \frac{1}{ad-bc} \begin{pmatrix} ad-bc & 0 \ 0 & ad-bc \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]Since this results in the identity matrix, \( A^{-1} \) is indeed the correct inverse.

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides vital information about the matrix, including whether the matrix is invertible. For a \(2 \times 2\) matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant, denoted as \(|A|\), is calculated as follows:\[ |A| = ad - bc \]The determinant can tell us if a matrix has an inverse. Specifically, for a matrix to be invertible, its determinant must not be zero \((|A| eq 0)\). If the determinant is zero, the matrix is said to be singular and has no inverse.

Inverse Matrix

An inverse matrix is another matrix that, when multiplied with the original matrix, results in the identity matrix. For a \(2 \times 2\) matrix \( A \), the inverse, denoted as \( A^{-1} \), exists only if the determinant \(|A| eq 0\). The formula to find the inverse of \( A \) is:\[ A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]This formula rearranges the elements of \( A \) and swaps the positions of \( a \) and \( d \), while \( b \) and \( c \) are negated. Finding the inverse is crucial for solving systems of linear equations and other applications in linear algebra.

2x2 Matrix

A \(2 \times 2\) matrix is a simple type of matrix, containing two rows and two columns. When dealing with \(2 \times 2\) matrices, such as\[ A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \]we can perform a variety of operations like addition, subtraction, multiplication, and finding determinants and inverses.

• Determinant: A measure of the matrix's invertibility.
• Inverse: Possible if the determinant is not zero.

Understanding the structure and properties of \(2 \times 2\) matrices is a foundational exercise in linear algebra.

Identity Matrix

The identity matrix is a special type of square matrix that serves as the multiplicative identity in matrix algebra. For a \(2 \times 2\) matrix, the identity matrix \( I \) is represented as follows:\[ I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]When any matrix is multiplied by the identity matrix, it remains unchanged:\[ A \cdot I = A \quad \text{and} \quad I \cdot A = A \]In the context of the inverse, verifying the inverse of a matrix involves confirming that the product of the matrix and its inverse equals the identity matrix: