Problem 44 - College Algebra with Modeling and Visualization

Question

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{llll}5 & -3\end{array}\right], \quad \quad \quad \quad \quad B=\left[\begin{array}{l}1 \\\3\end{array}\right]$$

Step-by-Step Solution

Answer

AB = -4; BA is not possible.

Step 1: Understanding Matrix Dimensions

Matrix \(A\) is a \(1 \times 2\) row matrix and Matrix \(B\) is a \(2 \times 1\) column matrix. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Therefore, we can find \(AB\) because \(A\) has 2 columns and \(B\) has 2 rows.

Step 2: Calculate AB

To find \(AB\), multiply the elements of the row in \(A\) by the corresponding elements of the column in \(B\) and then add them together. \[AB = \begin{bmatrix} 5 & -3 \end{bmatrix} \begin{bmatrix} 1 \ 3 \end{bmatrix} = 5 \times 1 + (-3) \times 3 = 5 - 9 = -4\]

Step 3: Analyzing the possibility of calculating BA

Matrix \(B\) is \(2 \times 1\) and Matrix \(A\) is \(1 \times 2\). To multiply \(B\) by \(A\), Matrix \(B\) would need as many columns as Matrix \(A\) has rows. Since \(B\) has 1 column and \(A\) has 1 row, we cannot perform the multiplication \(BA\).

Step 4: Conclusion

From our calculations, we can only find \(AB\), which results in \(-4\). The multiplication \(BA\) is not possible due to incompatible dimensions.

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial when it comes to understanding matrix multiplication. The dimensions of a matrix are described by the number of rows and columns it contains. For example, in our exercise, Matrix \(A\) is a \(1 \times 2\) matrix. This means it has 1 row and 2 columns. Meanwhile, Matrix \(B\) is a \(2 \times 1\) matrix, which means it has 2 rows and 1 column.

When you multiply matrices, you need to ensure that the number of columns in the first matrix matches the number of rows in the second matrix. In simpler terms, if Matrix \(A\) is an \(m \times n\) matrix, and Matrix \(B\) is a \(p \times q\) matrix, you can multiply them only if \(n = p\).

• This results in a new matrix with dimensions \(m \times q\).
• Understanding these dimensions is key to knowing which matrix multiplications are possible, as seen in the exercise where \(AB\) is possible but \(BA\) is not.

Row Matrix

A row matrix is a type of matrix with a single row. It is essential in matrix operations as it interacts differently compared to other matrix types. In our exercise, Matrix \(A\) is a row matrix of dimensions \(1 \times 2\).

Its form is as follows:

• It is written horizontally.
• Even though it occupies just one row, it can still influence the dimensions resulting from a multiplication with another matrix.

A row matrix can be multiplied with a column matrix of compatible size, resulting in a single number or sometimes another matrix. This uniqueness makes row matrices interesting and quite different from matrices with multiple rows.

Column Matrix

A column matrix consists of a single column of elements. It’s an essential concept in matrix multiplication, especially when paired with a row matrix. For instance, in our exercise, Matrix \(B\) is a column matrix of dimensions \(2 \times 1\).

Being single-column:

• This type of matrix is written vertically.
• During multiplication, it interacts with row matrices, resulting in simple multiplication to produce a scalar, as shown in the calculation of \(AB\).

Understanding column matrices helps in visualizing certain data sets and is useful for solving linear equations where each column can represent a distinct variable.

Matrix Operations

Matrix operations encompass all conceivable manipulations of matrices including addition, subtraction, and of course, multiplication, as addressed in our example.

For multiplication, consider this:

• Each element from the row of the first matrix multiplies with its corresponding element from the column of the second matrix.
• After performing these multiplications, add the results together.

This process is evident in our exercise:
- Multiplying Matrix \(A\), \([5, -3]\) with Matrix \(B\), \(\begin{bmatrix} 1 \ 3 \ \end{bmatrix}\), results in \(5 \times 1 + (-3) \times 3 = -4\).

This emphasizes how each operation follows defined rules that apply depending only on dimensions not the specific numbers. Understanding these operations ensures correct application in more complex problems.