Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show that the matrices are inverses of each other by showing that their product is the identity matrix \(I\). \(\left[\begin{array}{lll}3 & 2 & 3 \\ 2 & 2 & 1 \\ 2 & 1 & 1\end{array}\right]\) and \(\left[\begin{array}{rrr}-\frac{1}{3} & -\frac{1}{3} & \frac{4}{3} \\ 0 & 1 & -1 \\ \frac{2}{3} & -\frac{1}{3} & -\frac{2}{3}\end{array}\right]\)

Step-by-Step Solution

Answer

After calculating the products AB and BA using matrix multiplication, we get: AB = \(\left[\begin{array}{lll}1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1\end{array}\right]\) and BA = \(\left[\begin{array}{lll}1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1\end{array}\right]\) Both products are equal to the 3x3 identity matrix, so we can conclude that the given matrices A and B are inverses of each other.

Step 1: We are given two matrices: Matrix A = \(\left[\begin{array}{lll}3 & 2 & 3 \\\ 2 & 2 & 1 \\\ 2 & 1 & 1\end{array}\right]\) Matrix B = \(\left[\begin{array}{rrr}-\frac{1}{3} & -\frac{1}{3} & \frac{4}{3} \\\ 0 & 1 & -1 \\\ \frac{2}{3} & -\frac{1}{3} & -\frac{2}{3}\end{array}\right]\) #Step 2#: Calculate product AB

To calculate the product AB, we need to multiply each element in row 1 of matrix A by the corresponding element in column 1 of matrix B and add them up. Repeat this for all the other rows and columns. AB = \(\left[\begin{array}{lll}(3 \cdot -\frac{1}{3} + 2 \cdot 0 + 3 \cdot \frac{2}{3}) & (3 \cdot -\frac{1}{3} + 2 \cdot 1 + 3 \cdot -\frac{1}{3}) & (3 \cdot \frac{4}{3} + 2 \cdot -1 + 3 \cdot -\frac{2}{3}) \\\ (2 \cdot -\frac{1}{3} + 2 \cdot 1 + 1 \cdot \frac{2}{3}) & (2 \cdot -\frac{1}{3} + 2 \cdot 1 + 1 \cdot -\frac{1}{3}) & (2 \cdot \frac{4}{3} + 2 \cdot -1 + 1 \cdot -\frac{2}{3}) \\\ (2 \cdot -\frac{1}{3} + 1 \cdot 1 + 1 \cdot \frac{2}{3}) & (2 \cdot -\frac{1}{3} + 1 \cdot 1 + 1 \cdot -\frac{1}{3}) & (2 \cdot \frac{4}{3} + 1 \cdot -1 + 1 \cdot -\frac{2}{3})\end{array}\right]\) #Step 3#: Calculate product BA

Step 2: Now, we need to calculate the product BA by following the same process: BA = \(\left[\begin{array}{lll}(-\frac{1}{3} \cdot 3 + 0 \cdot 2 + \frac{2}{3} \cdot 2) & (-\frac{1}{3} \cdot 2 + 0 \cdot 2 + \frac{2}{3} \cdot 1) & (-\frac{1}{3} \cdot 3 + 0 \cdot 1 + \frac{2}{3} \cdot 1) \\\ (-\frac{1}{3} \cdot 3 + 1 \cdot 2 + -\frac{1}{3} \cdot 2) & (-\frac{1}{3} \cdot 2 + 1 \cdot 2 + -\frac{1}{3} \cdot 1) & (-\frac{1}{3} \cdot 3 + 1 \cdot 1 + -\frac{1}{3} \cdot 1) \\\ (\frac{4}{3} \cdot 3 + -1 \cdot 2 + -\frac{2}{3} \cdot 2) & (\frac{4}{3} \cdot 2 + -1 \cdot 2 + -\frac{2}{3} \cdot 1) & (\frac{4}{3} \cdot 3 + -1 \cdot 1 + -\frac{2}{3} \cdot 1)\end{array}\right]\) #Step 4#: Simplify AB and BA, compare results with the identity matrix

After simplifying the products AB and BA, we get: AB = \(\left[\begin{array}{lll}1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1\end{array}\right]\) BA = \(\left[\begin{array}{lll}1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1\end{array}\right]\) Since both AB and BA are equal to the 3x3 identity matrix, we can conclude that the given matrices A and B are inverses of each other.

Key Concepts

Inverse Matrices

Inverse matrices are essential in matrix theory and linear algebra. The concept of inverse matrices is similar to finding the reciprocal of a number. For a given non-zero number, the reciprocal is the number you multiply it by to get 1. In the matrix world, the inverse of a matrix is the matrix you multiply with to obtain the identity matrix.

An inverse matrix only exists for square matrices (matrices with the same number of rows and columns). Not all square matrices have inverses; only those with a non-zero determinant do. For instance, if you have matrix A, and there exists a matrix B such that their product is the identity matrix (AB = BA = I), then B is the inverse of A. You often denote this inverse as A^{-1}.

Finding the inverse of a matrix involves several methods, such as using Gaussian elimination or the adjugate method, but calculating inverses manually can sometimes become cumbersome. Thankfully, software like MATLAB can perform these operations efficiently.

Identity Matrix

The identity matrix is a fundamental concept in matrix algebra. It acts as the equivalent of the number 1 in numerical operations. When you multiply any matrix by the identity matrix, the original matrix remains unchanged, much like how multiplying a number by 1 leaves it the same.

The identity matrix is always a square matrix and contains ones along its main diagonal (from top left to bottom right) and zeros elsewhere. For instance, a 3x3 identity matrix looks like this:
\[ I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]

Understanding the identity matrix's role is crucial when exploring matrix inversions. In exercises surrounding inverse matrices, you can check your results by confirming that when you multiply the matrix by its inverse, you end up with the identity matrix.

Matrices Product

The product of matrices is a unique operation that combines rows from the first matrix with columns of the second. It's important to remember that matrix multiplication is not commutative, meaning that multiplying A by B does not necessarily yield the same result as multiplying B by A.

When performing matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. By doing so, each term from a row in the first matrix combines and sums up with corresponding terms from a column in the second matrix to form a new matrix.

Imagine calculating the product of two matrices. You multiply each element of a row by the corresponding element of a column, sum those products, and place the result in the corresponding position in the resulting matrix. This operation is vital for solving various mathematical problems and understanding more advanced matrix-related topics.

Matrix Operations

Matrix operations encompass a wide range of methods used to handle matrices. These include addition, subtraction, multiplication, and finding inverses, among others.

• Addition and Subtraction: For two matrices to be added or subtracted, they must be of the same dimension. You perform these operations element by element.
• Multiplication: As explored earlier, this involves row-to-column multiplication. Remember that it requires matching dimensions and is not commutative.
• Finding the Inverse: This involves creating or identifying a matrix which, when multiplied with the original, gives the identity matrix.
• Determinants: An important aspect when discussing matrix inverses. Only matrices with a non-zero determinant have inverses.

Understanding matrix operations is crucial for anyone working in areas of study like applied mathematics, engineering, computer science, and physics, where matrices often represent systems, transformations, and more.