Problem 17 - Algebra for College Students

Question

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{r} 2 \\ -7 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & -2 \\ 1 & 0 \\ -1 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

\(AB\) does not exist; \(BA = \begin{bmatrix} 20 \\ 2 \\ -30 \end{bmatrix}\).

Step 1: Determine if Product AB Exists

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \(A\) is \(2 \times 1\) and matrix \(B\) is \(3 \times 2\). Since the number of columns in \(A\) (1) does not equal the number of rows in \(B\) (3), the product \(AB\) does not exist.

Step 2: Determine if Product BA Exists

Similarly, for the product \(BA\) to exist, the number of columns in \(B\) (2) must equal the number of rows in \(A\) (2). Since these numbers are equal, the product \(BA\) does exist, and will result in a \(3 \times 1\) matrix.

Step 3: Calculate the Product BA

Matrix \(B\) is \(3 \times 2\), and matrix \(A\) is \(2 \times 1\). The product \(BA\) is calculated as follows:\[BA = \begin{bmatrix} 3 & -2 \ 1 & 0 \ -1 & 4 \end{bmatrix} \begin{bmatrix} 2 \ -7 \end{bmatrix} = \begin{bmatrix} (3)(2) + (-2)(-7) \ (1)(2) + (0)(-7) \ (-1)(2) + (4)(-7) \end{bmatrix} \]Simplifying each row, this becomes:\[\begin{bmatrix} 6 + 14 \ 2 + 0 \ -2 - 28 \end{bmatrix} = \begin{bmatrix} 20 \ 2 \ -30 \end{bmatrix}\]

Step 4: Present the Solution

The product \(AB\) does not exist due to incompatible dimensions, but the product \(BA\) does exist and is equal to the matrix:\[\begin{bmatrix} 20 \ 2 \ -30 \end{bmatrix}\]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when working with matrix multiplication. A matrix's dimensions are denoted as "rows by columns". For example, a matrix with 2 rows and 1 column is called a 2 \( \times \) 1 matrix. Similarly, a 3 \( \times \) 2 matrix has 3 rows and 2 columns.

Before performing any multiplication, it's important to check the dimensions of both matrices involved. Only some matrices can be multiplied together - they must be compatible in a specific way:

• The number of columns in the first matrix must match the number of rows in the second matrix.

In other words, if matrix \( A \) is \( m \times n \), it can only be multiplied by matrix \( B \) if \( B \) is \( n \times p \). This ensures all elements have corresponding factors to multiply together.

Existence of Matrix Product

Not all matrix products are possible, and the key factor is compatibility between their dimensions. Let's break this down with the provided exercise. Matrix \( A \) has dimensions of 2 \( \times \) 1, and matrix \( B \) has dimensions of 3 \( \times \) 2.

For the product \( AB \) to exist, the number of columns in \( A \) (which is 1) must equal the number of rows in \( B \) (which is 3).
Since these dimensions do not match, \( AB \) cannot be calculated.

Conversely, for the product \( BA \), the number of columns in \( B \) (2) must match the number of rows in \( A \) (2). These dimensions are equal, so the product \( BA \) is possible and results in a matrix of dimensions 3 \( \times \) 1.

Matrix Calculation Steps

To multiply matrices and obtain the product, follow a step-by-step approach. We will calculate the product \( BA \) based on the given matrices in the exercise:

1. **Identify the Compatible Pairs**: Matrix \( B \) is 3 \( \times \) 2 and matrix \( A \) is 2 \( \times \) 1. Since their respective dimensions align for multiplication, \( BA \) can be calculated and will yield a 3 \( \times \) 1 matrix.
2. **Multiply Each Element Appropriately**: Each row element of \( B \) is multiplied by each column element of \( A \) and summed up:

• First Row: \((3)(2) + (-2)(-7) = 6 + 14 = 20\)
• Second Row: \((1)(2) + (0)(-7) = 2 + 0 = 2\)
• Third Row: \((-1)(2) + (4)(-7) = -2 - 28 = -30\)

3. **Construct the Resulting Matrix**: Combine the sums to create a new matrix:

\[\begin{bmatrix} 20 \ 2 \ -30 \end{bmatrix}\]
Understanding these steps is key to mastering matrix multiplication and confidently determining the existence and result of a matrix product.