Problem 3 - College Algebra

Question

Find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative imerse of \(A\). $$A=\left[\begin{array}{rr}-4 & 0 \\\1 & 3\end{array}\right], \quad B=\left[\begin{array}{rr}-2 & 4 \\\0 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

After carrying out the matrix multiplications, neither \( AB \) nor \( BA \) equal the identity matrix, thus \( B \) is not the multiplicative inverse of \( A \).

Step 1: Multiplication AB

First, perform the matrix multiplication of \( A \) and \( B \). The element at the (i, j) position in the resulting matrix is obtained by taking the dot product of the ith row of \( A \) and the jth column of \( B \). Doing the calculations, the result is \( \[ \[ \begin{array}{rr}8 & -16 \0 & 3\end{array} \] \]

Step 2: Multiplication BA

Next, perform the matrix multiplication of \( B \) and \( A \). The element at the (i, j) position in the resulting matrix is obtained by taking the dot product of the ith row of \( B \) and the jth column of \( A \). Doing the calculations, the result is \( \[ \[ \begin{array}{rr}4 & 12 \1 & 3\end{array} \] \]

Step 3: Check if BA and AB equal identity

Now, check if matrices \( AB \) or \( BA \) are the identity matrix. Neither \( AB \) nor \( BA \) are the identity matrix. Therefore, \( B \) is not the multiplicative inverse of \( A \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, which combines two matrices to produce a third one. It's akin to the idea of dotting lines together such that specific rules are followed for the dimensions of the matrices and for how the numbers interact within the operation.

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. More formally, if you have a matrix A of size (m x n) and matrix B of size (n x p), their product, matrix C, will be of size (m x p).

The resulting matrix's entry in row i and column j, noted as C_ij, is calculated as the sum of the products of the corresponding elements from row i of the first matrix A and column j of the second matrix B. This sum of products is also known as the dot product, which we'll discuss more in another section.

From the given exercise:

Identity Matrix

In matrix algebra, the identity matrix plays a role similar to the number 1 in arithmetic. It is a square matrix with a special attribute: when any matrix is multiplied by the identity matrix, the original matrix is returned, untouched, as if by magic. This matrix is denoted by the symbol I, and it has 1's along its diagonal and 0’s elsewhere.

I comes in various sizes, but it always remains square; imagine a grid where 1's blaze a trail from the top-left corner to the bottom-right, leaving a trace of 0's in their wake. To check if a matrix B is the inverse of A, we multiply A by B and B by A, looking for the identity matrix as the result of both products. If we don't get the identity in both cases, then B is not the true inverse of A.

Dot Product in Matrices

The dot product, or scalar product, is a powerhouse operation that lies at the heart of matrix multiplication. It is a single number, obtained from two vectors of the same length, by multiplying corresponding entries and then summing those products. This is the crux of what happens within each entry of a matrix product.

In the context of our exercise, to get the entry of AB or BA at a particular position, say (1,1), you line up the components of row 1 of the first matrix with the matching elements of column 1 of the second matrix. Multiply pairs, sum it all up, and that's your golden number for this specific cell of the resulting matrix.

Understanding the dot product is paramount to grasping matrix multiplication—it's more than just combining numbers; it's a precise dance that adheres to strict algebraic choreography. Mastering it means you're on your way to understanding the deeper principles of linear algebra.