Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Inverse of a \(2 \times 2\) Matrix, use the formula on page 676 to find the inverse of the \(2 \times 2\) matrix (if it exists). $$\left[\begin{array}{rr} -\frac{1}{4} & -\frac{2}{3} \\ \frac{1}{3} & \frac{8}{9} \end{array}\right]$$

Step-by-Step Solution

Answer

The given matrix doesn't have an inverse because its determinant is zero.

Step 1: Identify the elements of the matrix

(a, b, c, d) = (-1/4, -2/3, 1/3, 8/9). These are the elements of the matrix given.

Step 2: Compute the determinant

First, we need to compute the determinant, det(A) = (ad - bc) = (-1/4 * 8/9) - (1/3*-2/3) = -2/9-(-2/9) = 0.

Step 3: Check the determinant

Since the determinant is equal to zero, the matrix does not have an inverse as per the definition of an invertible matrix. Therefore, no further steps are needed.

Key Concepts

Determinant of a Matrix

When dealing with matrices, one important concept is the determinant, especially in the context of determining if a matrix has an inverse. For a given 2x2 matrix:
\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]
The determinant is calculated using the formula \(ad - bc\). It is a scalar value that can tell us many things about the matrix.

• If the determinant is zero, the matrix is singular. This means it doesn't have an inverse.
• If the determinant is not zero, we've ensured that the matrix is invertible.

Calculating this value confirms whether we can proceed with matrix inversion or not. Understanding how this process works is crucial for managing matrices in algebra and beyond.

Matrix Inversion

Matrix inversion is the process of finding another matrix which, when multiplied with the original matrix, results in the identity matrix. This second matrix is known as the inverse.
Let’s explore the formula for inverting a 2x2 matrix if it exists. For matrix \(A\):
\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]
The inverse is calculated as:
\[\frac{1}{ad-bc} \begin{bmatrix}d & -b \-c & a\end{bmatrix}\]
Key points to remember:

• The determinant \((ad-bc)\) must be non-zero to use this formula. If zero, the matrix does not have an inverse.
• Swapping \(a\) and \(d\), and negating \(b\) and \(c\), is vital for forming the inverse.

This inversion technique is foundational in linear algebra, offering a way to "undo" matrix products.

2x2 Matrix Algebra

Understanding 2x2 matrices is a fundamental part of matrix algebra. They are matrices with two rows and two columns, forming the simplest non-trivial matrix cases. Let's summarize key ideas for students:

• Basic structure: A 2x2 matrix, \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), consists of four elements.
• Use in operations: These matrices can be added, subtracted, multiplied, and potentially inverted given a non-zero determinant.
• Importance of non-zero determinant: As seen in the example given, the determinant serves as a gateway to verify if inversion is feasible.
• Identity matrix: The result of multiplying a matrix by its inverse is the identity matrix \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

Equipping oneself with the ability to manage such operations forms a strong foundation in understanding larger, more complex matrix concepts later on.