Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Let \(A\) be a \(2 \times 2\) matrix given by $$ A=\left[\begin{array}{ll} x & 0 \\ 0 & y \end{array}\right] $$ Use the determinant of \(A\) to determine the conditions under which \(A^{-1}\) exists.

Step-by-Step Solution

Answer

The inverse of the 2x2 matrix A (denoted as \(A^{-1}\)) exists if and only if the product of \(x\) and \(y\) is not equal to zero, meaning that neither \(x\) nor \(y\) can equal zero.

Step 1: Understanding the Inverse of a Matrix

The inverse of a matrix 'A' denoted as \(A^{-1}\) only exists if 'A' is a non-singular matrix i.e., its determinant is not zero.

Step 2: Calculating the Determinant of the Matrix 'A'

To find the determinant of a 2x2 matrix, take the product of the elements on the main diagonal (top left to bottom right) and subtract the product of the elements on the other diagonal (top right to bottom left). The determinant for 'A' will be \(xy - 0*0\) which simplifies to \(xy\).

Step 3: Determining the Conditions

The inverse of the 2x2 matrix 'A' exists only when its determinant is not zero. The determinant of the matrix 'A' is \(xy\), which implies that as long as the product \(xy\) is not equal to zero, the inverse of matrix 'A' exists. This means that neither \(x\) nor \(y\) can be zero in order for \(A^{-1}\) to exist.

Key Concepts

Determinant

The determinant is a crucial number associated with a square matrix. For a given matrix, it helps us understand if certain operations, like finding an inverse, are possible. When dealing with a 2x2 matrix like matrix \( A \), the determinant can be calculated using a straightforward process.

For matrix \( A \), which is structured as \[A = \begin{bmatrix} x & 0 \ 0 & y \end{bmatrix}\] its determinant is found by multiplying the elements on its main diagonal, which are \(x\) and \(y\), and then subtracting the product of theother diagonal. For this matrix, the other diagonal consists of zeros, making the multiplication result zero as well. Thus, the determinant simplifies to \(xy\).

This calculation provides vital information about the matrix, especially when determining if an inverse exists.

Non-Singular Matrix

A matrix that has a non-zero determinant is known as a non-singular matrix. These matrices are quite significant because they possess certain desirable properties, like having an inverse.

If a matrix is non-singular, you can find another matrix such that their product yields the identity matrix. Just as important is understanding that a singular matrix, which has a determinant of zero, does not have this property, and thus, its inverse does not exist.

In our example of matrix \( A \), the condition for non-singularity is directly connected to the determinant being non-zero. The non-zero determinant indicates matrix \( A \) is non-singular, allowing it to possess an inverse.

Matrix Conditions

For a matrix to have an inverse, several conditions need to be satisfied. Understanding these conditions can guide us in determining potential matrix operations.

**Key Conditions for a 2x2 Matrix**

• The matrix needs to be square i.e., the same number of rows and columns.
• The determinant must not be zero. This ensures the matrix is non-singular.

Considering matrix \( A \), it's clear that as a 2x2 matrix, it is inherently square.
The determinant condition guides us further, ensuring \(xy eq 0\). This means both \(x\) and \(y\) must be non-zero for the inverse to exist.

These conditions are crucial in various mathematical and real-world applications where matrix inverses are involved.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices but provides excellent insight into matrix operations.
The structure of a 2x2 matrix is \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]This matrix comes with a convenient determinant formula: \(ad - bc\). Its simplicity makes it a perfect starting place for learning matrix determinants and inverses.

In our given problem, the simplicity of the 2x2 matrix makes calculating the determinant and checking the conditions for the inverse straightforward. Understanding the principles of a 2x2 matrix lays the foundation for dealing with larger and more complex matrices in future studies or applications.