Problem 41 - Differential Equations and Linear Algebra

Question

Consider the general \(2 \times 2\) matrix $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ and let \(\Delta=a_{11} a_{22}-a_{12} a_{21}\) with \(a_{11} \neq 0 .\) Show that if \(\Delta \neq 0\) $$ A^{-1}=\frac{1}{\Delta}\left[\begin{array}{rr} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{array}\right] $$ The quantity \(\Delta\) defined above is referred to as the determinant of \(A\). We will investigate determinants in more detail in the next chapter.

Step-by-Step Solution

Answer

In summary, given the general 2x2 matrix A and the determinant ∆ (with ∆ ≠ 0), we calculated the inverse of the matrix as \( A^{-1} = \frac{1}{\Delta} \left[ \begin{array}{rr} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{array} \right] \).

Step 1: Calculate the Determinant

First, we need to find the determinant of matrix A, denoted as ∆: \( \Delta = a_{11}a_{22} - a_{12}a_{21} \)

Step 2: Check if the determinant is non-zero

The given condition states that the determinant should not be equal to 0: \( \Delta \neq 0 \)

Step 3: Find the inverse matrix

Since the determinant is non-zero, the inverse of matrix A exists and can be calculated using the formula for a 2x2 matrix: \( A^{-1} = \frac{1}{\Delta} \left[ \begin{array}{rr} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{array} \right] \) Now we have shown that if the determinant is non-zero, the inverse of a general 2x2 matrix can be expressed in the given form.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special numerical value that is calculated from its elements. It is denoted by the symbol \( \Delta \) or sometimes as \( |A| \) when referring to matrix \( A \). For a general \( 2 \times 2 \) matrix \( A \), the determinant can be calculated using the formula:

\[ \Delta = a_{11}a_{22} - a_{12}a_{21} \]
This mathematical operation takes the product of the elements on the main diagonal (from the top-left to the bottom-right) and subtracts the product of the elements on the other diagonal (from the top-right to the bottom-left). It's a quick calculation, but its implications are vast, governing the properties and behaviors of the matrix in various applications, such as solving linear equations, inverting matrices, and even in more abstract fields like topology and geometry.

Non-Zero Determinant

When we talk about a non-zero determinant, we're essentially saying that the determinant of a matrix is not equal to zero \( (\Delta eq 0) \). This is an essential condition for a matrix to be invertible. The determinant provides information about the matrix, such as whether it is singular or non-singular. A non-zero determinant indicates that the matrix is non-singular, meaning it has an inverse. Furthermore, it confirms that the matrix describes a system with a unique solution and has a certain 'volume' in its transformation of space. Matrices with zero determinants are singular and do not have an inverse. They represent systems with either no solutions or infinitely many solutions and can flatten space into a lower dimension.

2x2 Matrix

A \( 2 \times 2 \) matrix is a square matrix that consists of two rows and two columns, usually represented as:

\[ A=\left[\begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right] \]
Each element in the matrix (denoted \( a_{ij} \)) represents a value at the intersection of the \( i^{th} \) row and the \( j^{th} \) column. The simplicity of a \( 2 \times 2 \) matrix allows for straightforward computations, yet it provides a foundation for understanding more complex matrix operations. Determinant and inverse calculations are exemplary initial steps in linear algebra, offering a glimpse into the behavior and characteristics of matrices on a manageable scale.

Inverse of Matrix

The inverse of a matrix \( A \) is another matrix, usually denoted as \( A^{-1} \) that, when multiplied with the original matrix, yields an identity matrix. However, not all matrices have an inverse. For a \( 2 \times 2 \) matrix, an inverse exists if and only if its determinant is non-zero. The formula for the inverse of a \( 2 \times 2 \) matrix is given as:

\[ A^{-1} = \frac{1}{\Delta}\left[\begin{array}{rr} a_{22} & -a_{12} \ -a_{21} & a_{11} \end{array}\right] \]
This formula rearranges the elements of the matrix \( A \) and scales them by the reciprocal of the determinant. The inverse matrix is crucial in solving systems of linear equations, as it provides a method to directly compute solutions. Additionally, it plays a significant role in various applications across mathematics, physics, and engineering.