Problem 46 - College Algebra

Question

If \(I\) is the multiplicative identity matrix of onder \(2,\) find \((I-A)^{-1}\) for the given matrix \(A\) $$\left[\begin{array}{rr}7 & -5 \\\\-4 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of (I-A) is \[ \left[\begin{array}{rr} \frac{1}{6} & \frac{5}{12} \\ \frac{1}{3} & \frac{1}{2} \end{array}\right] \

Step 1: Calculate I - A

First, subtract matrix A from the identity matrix I. The 2x2 identity matrix I is \[ \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right] \]So, \(I - A = \)\[ \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right] \] - \[ \left[\begin{array}{rr} 7 & -5 \\ -4 & 3 \end{array}\right] \]= \[ \left[\begin{array}{rr} 1-7 & 0-(-5) \\ 0-(-4) & 1-3 \end{array}\right] \]= \[ \left[\begin{array}{rr} -6 & 5 \\ 4 & -2 \end{array}\right] \

Step 2: Find Inverse of (I - A)

The formula for finding inverse of a 2x2 matrix\[ \left[\begin{array}{rr} a & b \\ c & d \end{array}\right] \]is \[ \frac{1}{ad-bc} * \left[\begin{array}{rr} d & -b \\ -c & a \end{array}\right] \].So, the inverse of \[ \left[\begin{array}{rr} -6 & 5 \\ 4 & -2 \end{array}\right] \]is \[ \frac{1}{(-6*(-2))-(5*4)} * \left[\begin{array}{rr} -2 & -5 \\ -4 & -6 \end{array}\right] \]= \[ \frac{1}{-12} * \left[\begin{array}{rr} -2 & -5 \\ -4 & -6 \end{array}\right] \]= \[ \left[\begin{array}{rr} \frac{1}{6} & \frac{5}{12} \\ \frac{1}{3} & \frac{1}{2} \end{array}\right] \

Key Concepts

Identity Matrix

In the world of matrices, an identity matrix is like the number 1 in regular arithmetic. It doesn't change any matrix it's multiplied with. A 2x2 identity matrix, denoted usually by the letter I, looks like this:\[\begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\]It's essentially a square matrix filled with zeros except for the main diagonal, which contains ones. This characteristic makes it the "do nothing" matrix in multiplication, meaning any matrix multiplied by an identity matrix remains unchanged. When you subtract a matrix from the identity matrix, like in the exercise you are examining, it creates a new matrix, highlighting the changes introduced by the matrix being subtracted.

2x2 Matrices

A 2x2 matrix is the simplest form of a square matrix, consisting of 2 rows and 2 columns. Here's an example matrix:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]When working with these matrices, you're handling four numbers in a rectangular format. This simplicity makes them relatively easy to manipulate, especially when learning foundational concepts like matrix addition, subtraction, and inverses. In practical applications, 2x2 matrices can be seen in transformations and rotations in graphics, among many other fields. They serve as a great introduction to understanding how larger matrices work due to their similar properties but with higher dimensions.

Matrix Subtraction

Matrix subtraction involves taking each element from one matrix and subtracting it from the corresponding element in another matrix. To be subtractable, matrices must have the same dimensions. Here's an example using two 2x2 matrices:\[\begin{bmatrix}1 & 2 \3 & 4\end{bmatrix} -\begin{bmatrix}5 & 6 \7 & 8\end{bmatrix} =\begin{bmatrix}1-5 & 2-6 \3-7 & 4-8\end{bmatrix} =\begin{bmatrix}-4 & -4 \-4 & -4\end{bmatrix}\]The goal is to adjust each element by the corresponding value from the second matrix. This operation highlights differences between matrices, just as matrix addition can show combined properties. Matrix subtraction is essential in various applications, including solving systems of linear equations and transformations.

Inverse Formula

Finding the inverse of a matrix is akin to finding a reciprocal in arithmetic. The inverse of a matrix, when multiplied with the original matrix, yields the identity matrix. For a 2x2 matrix \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], its inverse can be calculated using the formula:\[\frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]The expression \( ad-bc \) is called the determinant, which should not be zero for the inverse to exist. The determinant provides insight into the matrix's properties; if zero, the matrix is singular and non-invertible.This formula is significant because it offers a straightforward route to finding the inverse of a 2x2 matrix, usually involving rearranging and negating certain matrix elements. Understanding inverses is crucial for solving matrix equations and finding outliers in datasets.