Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{lll} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{array}\right] ; B=\left[\begin{array}{rrr} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

A and B are not inverses, as AB is not the identity matrix.

Step 1: Understand the Problem

To determine if matrices \(A\) and \(B\) are inverses of each other, we need to check if their product is the identity matrix. This involves calculating both \(AB\) and \(BA\). An identity matrix, \(I\), for a 3x3 matrix, is \(\left[\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right]\).

Step 2: Calculate AB

Multiply matrix \(A\) by matrix \(B\).\[AB = \left[\begin{array}{ccc}1 & 3 & 3 \1 & 4 & 3 \1 & 3 & 4\end{array}\right] \times \left[\begin{array}{ccc}7 & -3 & -3 \-1 & 1 & 0 \-1 & 0 & 1\end{array}\right] = \left[\begin{array}{ccc}0 & 1 & 0 \0 & 1 & 0 \0 & 0 & 1\end{array}\right]\]Each element of \(AB\) is computed by taking the dot product of corresponding row of \(A\) and column of \(B\).

Step 3: Calculate BA

Multiply matrix \(B\) by matrix \(A\).\[BA = \left[\begin{array}{ccc}7 & -3 & -3 \-1 & 1 & 0 \-1 & 0 & 1\end{array}\right] \times \left[\begin{array}{ccc}1 & 3 & 3 \1 & 4 & 3 \1 & 3 & 4\end{array}\right] = \left[\begin{array}{ccc}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{array}\right]\]Again, each element of \(BA\) is computed by taking the dot product of respective rows and columns.

Step 4: Compare Results with Identity Matrix

For matrices \(A\) and \(B\) to be inverses, both products \(AB\) and \(BA\) must equal the identity matrix. Here, \(AB = \left[\begin{array}{ccc} 0 & 1 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right]\) which is not the identity matrix, while \(BA = \left[\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right]\) which is the identity matrix. Since \(AB\) is not the identity matrix, \(A\) and \(B\) are not inverses.

Key Concepts

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce another matrix. It’s different from regular multiplication, as it requires dot products and is not commutative (meaning \(AB eq BA\) in general). To multiply a matrix \( A \) by a matrix \( B \), you need to consider these important points:

• The number of columns in \( A \) must be equal to the number of rows in \( B \).
• The resulting product will have the same number of rows as \( A \) and the same number of columns as \( B \).
• Each element in the product is calculated by taking the "dot product" of the respective row of \( A \) and column of \( B \).

These rules ensure that the matrices blend correctly, forming the correct dimensions and values for the resultant matrix. Matrix multiplication is a fundamental skill in many mathematical applications, such as solving systems of linear equations and transformations in geometry.

Identity Matrix

The identity matrix is a special kind of matrix. It functions like the number 1 in regular multiplication. When any matrix is multiplied by the identity matrix, it remains unchanged, just like any number multiplied by 1. For a square matrix of size \( n \times n \), the identity matrix \( I_n \) is denoted as:
\[I = \begin{bmatrix}1 & 0 & 0 & \cdots & 0 \0 & 1 & 0 & \cdots & 0 \0 & 0 & 1 & \cdots & 0 \\vdots & \vdots & \vdots & \ddots & \vdots \0 & 0 & 0 & \cdots & 1\end{bmatrix}\]

• Each diagonal element is 1.
• All off-diagonal elements are 0.

The identity matrix is crucial for confirming if two matrices are inverses. If both \( AB = I_n \) and \( BA = I_n \), then \( A \) and \( B \) are inverses of each other. This property is vital for matrix operations because it allows us to solve linear systems and perform various transformations efficiently.

Dot Product Calculation

The dot product is a vital component in matrix multiplication, calculating the values of individual elements in the product matrix. Here’s how to calculate it:

• Select a row from the first matrix.
• Select a column from the second matrix.
• Multiply each pair of corresponding elements from the row and column.
• Sum all the products to get a single number.

For example, if you have row \([a_1, a_2, a_3]\) and column \([b_1, b_2, b_3]\), the dot product is calculated as:
\[a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3\]
This calculation process is repeated to fill in each element of the resulting matrix. Understanding the dot product is crucial because it enables you to perform matrix multiplication accurately and efficiently, a necessary skill in various fields such as physics, computer graphics, and data analysis.