Problem 2 - Algebra and Trigonometry

Question

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rr} {-2} & {-1} \\ {-1} & {1} \end{array}\right], \quad B=\left[\begin{array}{ll} {1} & {1} \\ {1} & {2} \end{array}\right] $$

Step-by-Step Solution

Answer

B is not the multiplicative inverse of A because neither AB nor BA equals the identity matrix.

Step 1: Computing AB

First, compute the product AB. This is done by performing row-by-column multiplication. The top left element is found by \((-2)*1 + (-1)*1=-3\), the top right element by \((-2)*1 + (-1)*2=-4\), the bottom left element by \((-1)*1 + (1)*1=0\), and the bottom right element by\((-1)*1 + (1)*2=1 \). Therefore, AB is \(\left[\begin{array}{rr}{-3} & {-4} \ {0} & {1} \end{array}\right]\)

Step 2: Computing BA

Next, compute the product BA, again by performing row-by-column multiplication. The first element is found by \(1*(-2) + 1*(-1)=-3\), the second element by \(1*(-1) + 2*1=1\), the third element by \(1*(-2) + 2*(-1)=-4\), and the fourth element by \(1*(-1) + 2*1=1\). Therefore, BA is \(\left[\begin{array}{rr}{-3} & {1} \ {-4} & {1} \end{array}\right]\)

Step 3: Checking for the Identity Matrix

If B was the multiplicative inverse of A, then AB=BA would be the identity matrix. Since neither AB nor BA are the identity matrix, we can conclude B is not the multiplicative inverse of A.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in algebra that involves combining two matrices to produce a third matrix. Unlike element-wise multiplication, matrix multiplication combines information from both matrices in a structured manner. To perform this multiplication, we calculate the dot product of rows from the first matrix with columns from the second matrix. Here's a simplified explanation:

For instance, if you have a matrix A and want to multiply it by another matrix B, you'd take the first row of A and calculate the dot product with the first column of B to get the element in the first row and column of the resulting matrix. Then, you proceed to the next elements by working across the row of A and down the column of B until all elements of the resulting matrix are filled.

It's essential to note that the order of multiplication matters. Multiplying A by B can give a different result compared to multiplying B by A. This is because matrix multiplication is not commutative.

In our exercise, we multiplied matrices A and B to find AB and then reversed the order to find BA. The differences in the resulting matrices indicate that matrix multiplication order significantly affects the outcome.

Identity Matrix

The identity matrix, usually denoted as I, is a special square matrix that plays a similar role to the number 1 in regular multiplication. In an identity matrix, all the entries are 0 except for the diagonal from the upper left to the lower right corner, which contains 1s.

When any matrix A is multiplied by an identity matrix of the corresponding size, the result is the original matrix A. This unique property defines the identity matrix as the multiplicative identity of the matrix world.

When searching for the multiplicative inverse of a matrix, we aim to find another matrix such that when multiplied together, they result in the identity matrix. If the matrix B were the multiplicative inverse of A, then AB and BA would both equate to the identity matrix. However, in our exercise, neither product resulted in an identity matrix indicating that B is not the multiplicative inverse of A.

Row-by-Column Multiplication

Row-by-column multiplication is the specific method used to multiply matrices, and it is based on the dot product of vectors. This technique requires a systematic approach where each element of the resulting matrix is calculated by multiplying corresponding elements from the row of the first matrix and the column of the second matrix, then adding them together.

To illustrate, consider the element located in the ith row and jth column of the product matrix. This element is computed by taking the ith row from the first matrix and the jth column from the second matrix, multiplying them element-wise, and summing the products.

In the given exercise, we applied row-by-column multiplication to find the products AB and BA. Each element in the product matrices reflects the sum of these dot products, which conveys the intertwined relationship between the data from the original matrices. It's crucial to align the rows and columns correctly and perform the calculations with precision to ensure an accurate product matrix.