Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Let \(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & -1\end{array}\right] .\) Show that \(A^{3}=I_{3},\) and use this result to find the inverse of \(A\)

Step-by-Step Solution

Answer

The inverse of \(A\) is \(A^2\).

Step 1: Calculate \(A^2\)

Multiply matrix \(A\) by itself to find \(A^2\). Given:\[ A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & -1 \ 0 & 1 & -1 \end{bmatrix} \]Calculate \(A^2 = A \cdot A\):\[ A^2 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & -1 \ 0 & 1 & -1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & -1 \ 0 & 1 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & -1 & 1 \ 0 & -1 & 0 \end{bmatrix} \]

Step 2: Calculate \(A^3\)

Multiply the result from Step 1 by \(A\) to find \(A^3\):\[ A^3 = A^2 \cdot A = \begin{bmatrix} 1 & 0 & 0 \ 0 & -1 & 1 \ 0 & -1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & -1 \ 0 & 1 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]Thus, \(A^3 = I_3\), the identity matrix.

Step 3: Find the Inverse of \(A\)

Since \(A^3 = I_3\), this means \(A \, A \, A = I_3\). Thus, \(A^2\) is the inverse of \(A\).\[ A^{-1} = A^2 = \begin{bmatrix} 1 & 0 & 0 \ 0 & -1 & 1 \ 0 & -1 & 0 \end{bmatrix} \]

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying two matrices to create a new matrix. It’s essential because it combines information from both matrices in a structured way.
To multiply matrices, follow these key points:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• The resulting matrix has the dimensions of the number of rows from the first matrix by the number of columns from the second matrix.

Each element in the resulting matrix is calculated by taking the corresponding row from the first matrix and column from the second matrix, multiplying each pair of corresponding elements, and summing those products.

Keep this in mind while multiplying matrices, as it ensures you adhere to the mathematical rules and obtain correct results. For example, when finding \(A^3\), you multiply \(A^2\) by \(A\), paying attention to the rows and columns just as described. This is what you'll use to reach back to the concept of the identity matrix.

Identity Matrix

The identity matrix is the multiplicative identity in matrix algebra, similar to how the number 1 is the multiplicative identity in arithmetic.
An identity matrix is a square matrix, often denoted as \(I_n\), where all the elements on the main diagonal are 1, and all other elements are 0.

This matrix has a special property: for any matrix \(A\) which can multiply it, \(A\cdot I = I\cdot A = A\). It's like multiplying a number by 1.
Remember that in the exercise, achieving \(A^3 = I_3\) implies you have cycled back to the equivalent of the "original" version of \(A\), emphasizing that the operations didn’t change \(A\)'s inherent characteristics. Recognizing when you reach or should expect an identity matrix helps in determining if you’ve performed the operation correctly.

Inverse Matrix

An inverse matrix is akin to the reciprocal in regular arithmetic. For a square matrix \(A\), its inverse \(A^{-1}\) is such that when you multiply them, you get the identity matrix: \(A\cdot A^{-1} = I\). Not every matrix has an inverse, but when it does, it allows you to "undo" the effect of multiplying by \(A\).
To understand inverse matrices deeply:

• A matrix must be square to have an inverse.
• The determinant of the matrix should not be zero.

The exercise demonstrated that \(A^2\) is the inverse of \(A\), since \(A\cdot A^2 = I_3\). This highlights the cyclical property acting as a check to ensure the overall matrix operation results in the identity matrix. Recognizing and finding an inverse is crucial because it allows one to solve matrix equations and understand transformations within linear algebra.