Problem 19 - Algebra for College Students

Question

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{r} 3 \\ -4 \\ 2 \end{array}\right], \quad B=\left[\begin{array}{ll} 3 & -4 \end{array}\right] $$

Step-by-Step Solution

Answer

AB exists and is \( \begin{bmatrix} 9 & -12 \\ -12 & 16 \\ 6 & -8 \end{bmatrix} \); BA does not exist.

Step 1: Understand Matrix Dimensions

For matrix multiplication to be possible between two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix A is a 3x1 matrix and matrix B is a 1x2 matrix.

Step 2: Check for Possibility of Multiplying AB

To multiply matrices A and B, the number of columns in A (which is 1) must match the number of rows in B (which is also 1). Therefore, the product AB is possible and will result in a 3x2 matrix.

Step 3: Compute AB

Multiply each element of the matrix A by each element in the matrix B: \[AB = \begin{bmatrix} 3 \cdot 3 & 3 \cdot (-4) \ -4 \cdot 3 & -4 \cdot (-4) \ 2 \cdot 3 & 2 \cdot (-4) \end{bmatrix} = \begin{bmatrix} 9 & -12 \ -12 & 16 \ 6 & -8 \end{bmatrix}\]

Step 4: Check for Possibility of Multiplying BA

To multiply B and A, the number of columns in B (which is 2) must equal the number of rows in A (which is 3). Since this is not the case, the product BA does not exist.

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial to understanding matrix multiplication because they define the shape and size of a matrix. A matrix is typically written in the format of 'rows by columns.' For example, a matrix with 3 rows and 1 column is described as a 3x1 matrix.
In matrix operations, especially multiplication, matrix dimensions dictate whether two matrices can indeed be multiplied together.
For two matrices, such as A and B, to be multiplicable:

• The number of columns in Matrix A must be equal to the number of rows in Matrix B.

This compatibility condition is vital. If not met, the multiplication cannot happen, and trying to multiply would result in an undefined operation. Understanding matrix dimensions helps ensure that the elements are aligned correctly when performing operations like multiplication.

Matrix Product

The matrix product, or matrix multiplication, involves two matrices and combines their elements according to specific rules. Each element from the resulting matrix comes from the sum of the products of elements from the rows of one matrix and the columns of the other.
Here are some essentials about the matrix product:

• The order of multiplication matters. Hence, A multiplied by B (AB) is usually different from B multiplied by A (BA).
• If matrix A is of size m x n and matrix B is of n x p, the resulting AB will be a matrix of size m x p.

In our exercise, matrix A is 3x1, and B is 1x2. This means we can compute AB, resulting in a 3x2 matrix because the inner dimensions (1 in A and 1 in B) match. The multiplication process is facilitated by calculating entries via row-to-column product sums, as demonstrated in the given exercise.

Scalar Multiplication

Scalar multiplication is not the same as matrix multiplication, but learning about it can help build intuition around matrix operations. Scalar multiplication involves multiplying every element in a matrix by a single number (the scalar).
This operation scales the entries of the matrix, hence the term 'scalar multiplication.' Here’s a simple way to understand it:

• Each entry in the matrix is multiplied individually by the scalar value.
• This operation does not change the dimensions of the matrix.

While not directly used in our exercise involving matrices A and B, scalar multiplication is a fundamental concept in linear algebra that shares ideas with broader matrix operations. It marks an essential shift from scalar arithmetic to the manipulation of arrayed data in matrices.