Problem 48 - College Algebra with Modeling and Visualization

Question

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrrr}1 & -1 & 3 & -2 \\\1 & 0 & 3 & 4 \\\2 & -2 & 0 & 8 \end{array}\right], \quad B=\left[\begin{array}{rr}1 & -1 \\\0 & 5 \\\2 & 3 \\\\-5 & 4\end{array}\right]$$

Step-by-Step Solution

Answer

\(AB = \begin{bmatrix} 25 & -14 \\ -11 & 23 \\ -38 & 16 \end{bmatrix}\); \(BA\) not possible.

Step 1: Determine Compatibility for A B

Matrix multiplication is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. Matrix \(A\) has 4 columns, and matrix \(B\) has 4 rows. Thus, the multiplication \(AB\) is possible.

Step 2: Calculate A B

To calculate \(AB\), multiply each element of the rows of \(A\) by each element of the columns of \(B\) and sum the products for each element in the resulting matrix. \[AB = \begin{bmatrix} 1\cdot1 + (-1)\cdot0 + 3\cdot2 + (-2)\cdot(-5) & 1\cdot(-1) + (-1)\cdot5 + 3\cdot3 + (-2)\cdot4 \ 1\cdot1 + 0\cdot0 + 3\cdot2 + 4\cdot(-5) & 1\cdot(-1) + 0\cdot5 + 3\cdot3 + 4\cdot4 \ 2\cdot1 + (-2)\cdot0 + 0\cdot2 + 8\cdot(-5) & 2\cdot(-1) + (-2)\cdot5 + 0\cdot3 + 8\cdot4 \end{bmatrix} = \begin{bmatrix} 25 & -14 \ -11 & 23 \ -38 & 16 \end{bmatrix} \]

Step 3: Determine Compatibility for B A

The multiplication \(BA\) requires matrix \(B\) and matrix \(A\) to have compatible dimensions: the number of columns in \(B\) must equal the number of rows in \(A\). Since \(B\) has 2 columns and \(A\) has 3 rows, \(BA\) is not possible.

Step 4: Final Results

The product \(AB\) is a \(3 \times 2\) matrix: \( \begin{bmatrix} 25 & -14 \ -11 & 23 \ -38 & 16 \end{bmatrix} \). The product \(BA\) is not defined as the dimensions are incompatible.

Key Concepts

Linear Algebra

Linear algebra is a crucial branch of mathematics that deals with vectors, vector spaces, and matrices. It provides a framework for understanding systems of linear equations, transformations, and more. In the context of matrices, linear algebra is fundamental because it allows us to systematically perform operations like addition, subtraction, and multiplication on matrices.

Matrix multiplication is a central concept in linear algebra and it requires understanding how the individual components (i.e., rows and columns) of matrices interact. Through multiplication, matrices can represent complex transformations and systems compactly. This makes linear algebra immensely powerful in various fields including computer science, physics, and engineering.

Typically, when multiplying matrices, you'll employ the dot product of rows and columns to compute the elements of the resulting matrix. This operation is critical in many real-world applications, including graphics transformations and optimization problems.

Matrix Compatibility

Matrix compatibility is essential to ensure matrix multiplication can be performed. It refers to the specific requirement that must be met for the two matrices to be multiplied together. For any two matrices, say Matrix \(A\) and Matrix \(B\), the rule is:

• The number of columns in Matrix \(A\) must equal the number of rows in Matrix \(B\).

This specific requirement is crucial because each element of the resulting matrix arises from the interactions between the rows of the first matrix and the columns of the second matrix.

Consider the exercise given, where Matrix \(A\) is a \(3 \times 4\) matrix and Matrix \(B\) is a \(4 \times 2\) matrix. The compatibility check confirms that multiplication \(AB\) is possible because Matrix \(A\) has the same number of columns as Matrix \(B\) has rows, allowing the calculation to proceed.

Matrix Dimensions

Matrix dimensions play a vital role in determining both the feasibility of operations and the nature of the output matrix. Dimensions are denoted by rows and columns, e.g., a matrix with 3 rows and 2 columns is written as \(3 \times 2\).

When doing matrix multiplication, the dimensions of the resulting matrix are determined by the following rule:

• If matrix \(A\) has dimensions \(m \times n\) and matrix \(B\) has dimensions \(n \times p\), the resulting matrix \(AB\) will have dimensions \(m \times p\).

In the exercise, considering Matrix \(A\) is \(3 \times 4\) and Matrix \(B\) is \(4 \times 2\), the final product matrix \(AB\) should have dimensions \(3 \times 2\).

This rule stems from the interaction between rows and columns during multiplication, where each entry in the resulting matrix is computed based on a specific row from the first matrix and a specific column from the second matrix.