Problem 45 - College Algebra

Question

If \(I\) is the multiplicative identity matrix of onder \(2,\) find \((I-A)^{-1}\) for the given matrix \(A\) $$\left[\begin{array}{rr}8 & -5 \\\\-3 & 2\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of \( (I - A) \) is \(\left[\begin{array}{rr}-8 & 5 \\ 3 & -2\end{array}\right]\).

Step 1: Subtraction

Subtract matrix \(A\) from the identity matrix \(I\).

Step 2: Determinant Calculation

Calculate the determinant (denoted as det) of the resultant matrix, using the formula det = \(ad - bc\) where \(a, b, c, d\) are matrix elements.

Step 3: Adjugate Matrix Creation

Form the adjugate of the resultant matrix (swap the diagonal elements, and multiply the off-diagonal elements by -1).

Step 4: Inverse Matrix Calculation

Calculate the inverse by dividing the adjugate matrix by the determinant. This gives the required output.

Key Concepts

Identity Matrix

In linear algebra, the identity matrix, often denoted as I, plays a crucial role akin to the number 1 in regular arithmetic; it effectively leaves any matrix unchanged when used in multiplication. For instance, if you have a square matrix A and multiply it by the identity matrix I of the same order, the result will be A itself (i.e., AI = IA = A).

The identity matrix has a simple structure, where all the elements on the main diagonal are 1, and all other elements are 0. For a 2x2 matrix, the identity matrix I looks like this:
\[ I = \begin{bmatrix} 1 & 0 \ 0 & 1 \ end{bmatrix} \]

Matrix Subtraction

Subtracting one matrix from another is straightforward: you subtract the corresponding elements of one matrix from the other. However, this operation is only defined when both matrices have the same dimensions.

For the given problem, we start by subtracting matrix A from the identity matrix I of the same order, which yields:
\[ (I - A) = \begin{bmatrix} 1 - 8 & 0 - (-5) \ 0 - (-3) & 1 - 2 \ end{bmatrix} \]
The resulting matrix from this operation will serve as the base for finding the inverse in further steps.

Determinant of a Matrix

The determinant of a matrix is a scalar value that is a useful property in linear algebra, particularly when dealing with square matrices. It can tell you about certain characteristics of the matrix, such as whether it has an inverse or its scaling factor on volume when it represents a transformation.

The determinant for a 2x2 matrix A, with elements \( a, b, c, d \), is calculated as \[ det(A) = ad - bc \]. In the context of our problem, after subtracting A from I, computing the determinant of the resultant matrix is a critical step that will subsequently allow us to find the inverse of (I - A), provided that the determinant is not zero (as a zero determinant implies that the matrix is non-invertible).

Adjugate Matrix

The adjugate of a matrix, sometimes called the adjoint matrix, is closely related to the inverse of a matrix. For a 2x2 matrix, the process to find the adjugate is pretty simple: you swap the elements on the main diagonal and change the signs of the off-diagonal elements. Effectively, the adjugate of a matrix A with elements \( a, b, c, d \) is:
\[ adj(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \].

In our context, finding the adjugate matrix of \( (I - A) \) is necessary to compute its inverse by multiplying the adjugate by 1/det(I - A), assuming the determinant is non-zero. If the determinant is zero, the matrix does not have an inverse, which indicates a singular matrix.