Problem 10 - Algebra for College Students

Question

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{ll} 3 & -4 \\ 6 & -8 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix does not have a multiplicative inverse since its determinant is 0.

Step 1: Understand the Problem

We are asked to find the multiplicative inverse of a given 2x2 matrix, if it exists. The multiplicative inverse of a matrix \( A \) is another matrix \( A^{-1} \) such that \( A \times A^{-1} = I \), where \( I \) is the identity matrix.

Step 2: Calculate the Determinant

Before finding the inverse of the matrix \( \left[\begin{array}{cc} 3 & -4 \ 6 & -8 \end{array}\right] \), we need to calculate its determinant. The determinant of a 2x2 matrix \( \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \) is given by \( ad - bc \). Substitute the values \( a = 3 \), \( b = -4 \), \( c = 6 \), and \( d = -8 \) into the formula: Determinant = \( 3(-8) - (-4)(6) = -24 + 24 = 0 \).

Step 3: Check the Determinant

Since the determinant of the matrix is 0, it means that the matrix is singular and does not have an inverse.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that gives us important information about the matrix. For a 2x2 matrix, the determinant can be calculated using a simple formula. If we have a matrix like \( \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \), the determinant is given by \( ad - bc \). This formula is easy to remember and apply.

The value of the determinant tells us a lot about the matrix. If the determinant is zero, it indicates that the matrix is singular, meaning it does not have an inverse. On the other hand, if the determinant is not zero, the matrix does have an inverse. In solving problems involving matrices, calculating the determinant is a crucial first step. It helps us decide whether we can go ahead with finding the inverse or if there is no inverse at all.

To put it simply, calculating the determinant is like checking whether the matrix is 'invertible' or not, which can save us time and effort when dealing with matrices.

Singular Matrix

A singular matrix is a square matrix that does not have an inverse. The primary reason for a matrix being singular is when its determinant is zero.

In matrix terms, being singular means the row vectors of the matrix are linearly dependent; one row can be written as a combination of others. This dependency causes the determinant to be zero, and hence, an inverse cannot be calculated.

It is important to recognize a singular matrix early on, especially in mathematics and engineering problems. Knowing a matrix is singular can inform us about potential solutions or simplify complex systems by understanding that they might not have unique solutions.

Encountering a singular matrix means we should be cautious in our approach, as traditional methods that rely on inverses won't work. Instead, we need alternative methods like employing the concept of pseudoinverse or other numerical techniques when dealing with such matrices.

2x2 Matrix Inversion

Finding the inverse of a 2x2 matrix is a straightforward process, provided the matrix is not singular. For the matrix \( \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \), the inverse, denoted as \( \left[\begin{array}{cc} e & f \ g & h \end{array}\right] \), can be found using the formula:
\[ A^{-1} = \frac{1}{ad - bc} \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right] \]
This formula uses the determinant \( ad - bc \) as a divisor. This means that the determinant must be non-zero (the matrix is non-singular) to find an inverse.

When solving problems, once the determinant is confirmed to be non-zero, we can proceed with finding the inverse using this formula. The calculation involves adjusting the elements of the original matrix, swapping the positions of \( a \) and \( d \), and changing the signs of \( b \) and \( c \). Then, each element is multiplied by the reciprocal of the determinant.

This simple formula makes inverting 2x2 matrices an easy and quick task, enabling us to solve systems of equations or transform data efficiently.