Problem 35 - College Algebra with Modeling and Visualization

Question

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rr}3 & -1 \\\1 & 0 \\\\-2 & -4\end{array}\right], \quad B=\left[\begin{array}{rrr}-2 & 5 & -3 \\\9 & -7 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The product \( AB \) is defined with a result of \( \begin{bmatrix} -15 & 22 & -9 \\ -2 & 5 & -3 \\ -32 & 18 & 6 \end{bmatrix} \), but \( BA \) is not defined.

Step 1: Dimensions of A and B

Matrix \( A \) is a \( 3 \times 2 \) matrix, and matrix \( B \) is a \( 2 \times 3 \) matrix.

Step 2: Determining if AB Exists

For the product \( AB \) to be defined, the number of columns in \( A \) must equal the number of rows in \( B \). Since \( A \) has 2 columns and \( B \) has 2 rows, \( AB \) is defined. The resulting matrix will be \( 3 \times 3 \).

Step 3: Calculating AB

Calculate the product \( A B \) by taking the dot product of the rows of \( A \) with the columns of \( B \):\[AB = \begin{bmatrix}3(-2) + (-1)(9) & 3(5) + (-1)(-7) & 3(-3) + (-1)(0) \1(-2) + 0(9) & 1(5) + 0(-7) & 1(-3) + 0(0) \-2(-2) + (-4)(9) & -2(5) + (-4)(-7) & -2(-3) + (-4)(0)\end{bmatrix}\]This results in:\[AB = \begin{bmatrix}-6 - 9 & 15 + 7 & -9 + 0 \-2 + 0 & 5 + 0 & -3 + 0 \4 - 36 & -10 + 28 & 6 + 0\end{bmatrix} = \begin{bmatrix}-15 & 22 & -9 \-2 & 5 & -3 \-32 & 18 & 6\end{bmatrix}\]

Step 4: Determining if BA Exists

For the product \( BA \) to be defined, the number of columns in \( B \) must equal the number of rows in \( A \). Since \( B \) has 3 columns and \( A \) has 3 rows, \( BA \) is not defined.

Key Concepts

Understanding Matrix Dimensions

When dealing with matrices, the first step is to comprehend their dimensions, which are always given in the form of rows by columns. This is crucial because it helps determine the feasibility of operations such as addition, subtraction, and importantly, multiplication. For example, consider matrix \( A \) with dimensions \( 3 \times 2 \), which means it has 3 rows and 2 columns. Conversely, matrix \( B \) has dimensions \( 2 \times 3 \), having 2 rows and 3 columns.

Knowing these dimensions sets the ground for determining whether a matrix product can be executed, as there are specific rules regarding which dimensions are compatible. Remember, while working with matrices, always specify dimensions to avoid operation errors.

The Dot Product in Matrix Multiplication

The concept of the dot product is integral to matrix multiplication. To compute the product of two matrices, we take the dot product of the entries in the rows of the first matrix with the entries in the columns of the second.

Let's consider finding the dot product of row 1 from matrix \( A \) and column 1 from matrix \( B \):

• Multiply corresponding elements from the two: \(3 \times -2\) and \((-1) \times 9\).
• Then sum these results: \(-6 - 9 = -15\).

This resulting sum becomes the element at position (1, 1) in the new matrix. This process is repeated for each combination of row and column across the two matrices to fully construct the product matrix.

Matrix Product Existence

Before trying to find the product of two matrices, we need to determine if this product can actually exist. This is known as checking the matrix product existence condition. For two matrices to be multiplied, the number of columns in the first matrix must exactly match the number of rows in the second matrix.

In our specific example:

• To find \( AB \), matrix \( A \) has 2 columns and matrix \( B \) has 2 rows, so the product exists and results in a \( 3 \times 3 \) matrix.
• To find \( BA \), however, matrix \( B \) has 3 columns and matrix \( A \) has 3 rows. Here, the number of columns in \( B \) matches the number of rows in \( A \). Surprisingly, this might seem enough to form a product, but it is not possible to execute \( BA \) because the resultant matrix does not align dimensionally as the inner dimensions don't match.

Always verify this condition first to save time and avoid errors while performing matrix arithmetic.