Problem 6 - Algebra for College Students

Question

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{ll} 1 & -2 \\ 4 & -3 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix has a multiplicative inverse: \(\begin{bmatrix} -\frac{3}{5} & \frac{2}{5} \\ -\frac{4}{5} & \frac{1}{5} \end{bmatrix}\).

Step 1: Identify the Matrix

First, we identify the matrix given in the exercise:\[A = \begin{bmatrix} 1 & -2 \ 4 & -3 \end{bmatrix}\]We need to find if a multiplicative inverse exists for this matrix.

Step 2: Find the Determinant

The first step in determining the existence of an inverse is to calculate the determinant of matrix \(A\). For a 2x2 matrix, the determinant \(det(A)\) is given by:\[det(A) = ad - bc = 1(-3) - (-2)(4)\]Calculate it:\[det(A) = -3 + 8 = 5\]Since the determinant is not zero, an inverse exists.

Step 3: Apply Formula for Inverse

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse \(A^{-1}\) is given by:\[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]Substitute the values \(a = 1, b = -2, c = 4, d = -3\) into the formula:\[A^{-1} = \frac{1}{5}\begin{bmatrix} -3 & 2 \ -4 & 1 \end{bmatrix}\]

Step 4: Calculate the Inverse Matrix

Multiply each element of the matrix by \(\frac{1}{5}\):\[A^{-1} = \begin{bmatrix} -\frac{3}{5} & \frac{2}{5} \ -\frac{4}{5} & \frac{1}{5} \end{bmatrix}\]This gives us the inverse of the original matrix.

Key Concepts

Determinant of Matrix

Understanding the concept of a determinant is essential when working with matrices, especially if you're aiming to find their inverses. The determinant of a matrix can be thought of as a special number that provides insight into various properties of the matrix. For instance, it tells us whether a matrix is invertible or not. For a 2x2 matrix given by: i = \begin{bmatrix} a & b \ c & d \end{bmatrix}, the determinant is calculated as:\[det(A) = ad - bc\]This formula is relatively straightforward for a 2x2 matrix. Simply multiply the two diagonal numbers and subtract the product of the other diagonal. A determinant equal to zero means the matrix does not have an inverse. On the other hand, if it's non-zero, the matrix is invertible. Interpreting the determinant becomes second nature with practice. It's fundamentally vital for finding the inverse.

Multiplicative Inverse

The multiplicative inverse of a matrix, when it exists, transforms the matrix back to its identity form when multiplied together. It works much like finding the reciprocal of a number, but instead of numbers, we deal with matrices. For a square matrix \(A\), the multiplicative inverse \(A^{-1}\) is defined such that:\[A \cdot A^{-1} = I\]Here, \(I\) is the identity matrix, which is the matrix equivalent of the number one in multiplication. It means that any matrix multiplied by the identity matrix returns itself.To find the multiplicative inverse of a 2x2 matrix, you first need to ensure the determinant is non-zero. After confirming this, you apply a formula specific to 2x2 matrices, utilizing the terms of the original matrix.This property is crucial in solving linear equations, where finding a multiplicative inverse can simplify the process of obtaining solutions significantly.

2x2 Matrix Inversion

Inverting a 2x2 matrix is quite a systematic process, which involves a few straightforward steps once you get the hang of it. Begin by ensuring the matrix is square (it has the same number of rows as columns) and then compute its determinant to check if the inverse exists. For the matrix form:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]The inverse \(A^{-1}\) can be found using:\[A^{-1} = \frac{1}{det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]You'll swap the positions of \(a\) and \(d\), change the signs of \(b\) and \(c\), and scale the entire matrix by the reciprocal of the determinant.This method works because it effectively reverses the transformation represented by the original matrix, offering a powerful tool for solving system of equations, among other things. Regular practice will make matrix inversion feel like second nature.