Problem 7 - College Algebra

Question

find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right] \quad B=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

Yes, B is the multiplicative inverse of A, as both AB and BA resulted in Identity matrices.

Step 1: Multiply A and B Matrices

Firstly, determine the product AB. Matrix multiplication is done by performing dot product of the rows of the first matrix with the columns of the second matrix. So, \(A B= \left[\begin{array}{lll} 0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 0 \end{array}\right] \times \left[\begin{array}{lll} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{array}\right] = \left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right] \)

Step 2: Multiply B and A Matrices

Next, we determine the product BA, executing the same process. So, \(B A= \left[\begin{array}{lll} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{array}\right] \times \left[\begin{array}{lll} 0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 0 \end{array}\right] = \left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right]\)

Step 3: Verify Multiplicative Inverse

Finally, we verify if B is the multiplicative inverse of A. If both products AB and BA result in the Identity matrix (3x3 matrix with 1's on the main diagonal and elsewhere 0's), then B is indeed the multiplicative inverse of A. As both the products resulted in an Identity matrix, so yes, B is the multiplicative inverse of A.

Key Concepts

Multiplicative Inverse

The multiplicative inverse of a matrix is a key concept in linear algebra. If a matrix \( B \) is the multiplicative inverse of matrix \( A \), multiplying \( A \) by \( B \) yields the identity matrix. This means:

• \( A \times B = I \)
• \( B \times A = I \)

Here, \( I \) represents the identity matrix. If both products of matrices \( A \) and \( B \) result in the identity matrix, then \( B \) is a genuine multiplicative inverse of \( A \).
For example, in the exercise, matrices \( A \) and \( B \) were multiplied in both orders and each product was an identity matrix. Therefore, \( B \) is indeed the multiplicative inverse of \( A \). Understanding this concept helps in solving systems of linear equations and finding solutions for matrix equations.

Identity Matrix

The identity matrix plays an essential role in matrix operations, similar to the number '1' in multiplication for regular numbers. An identity matrix has distinct characteristics:

• It is a square matrix with ones (1's) on the main diagonal.
• All other elements are zeros (0's).

For instance, a 3x3 identity matrix looks like this:\[I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]In matrix multiplication, the identity matrix behaves neutrally:

• \( A \times I = A \)
• \( I \times A = A \)

This property is very useful when verifying inverses. In the exercise, multiplication results in the identity matrix when matrices \( A \) and \( B \) are involved, confirming that one is the inverse of the other.

Dot Product

The dot product is a fundamental concept in matrix multiplication.
When calculating the product of two matrices, the dot product is used for corresponding row and column elements.
Here’s the process:

• Take a row from the first matrix.
• Take a column from the second matrix.
• Multiply each element from the row with the corresponding element from the column.
• Add the results of these multiplications together to get a single entry for the product matrix.

This operation is repeated for each row-column pair in the matrices. As a result, dot products help produce a new matrix containing all necessary computations.
For instance, in our exercise, the matrices \( A \) and \( B \) were multiplied using dot products to determine the products \( AB \) and \( BA \). Understanding dot products ensures you can effectively execute matrix multiplication with accuracy and confidence.