Problem 49 - Algebra 2

Question

Determine whether each matrix product is defined. If so, state the dimensions of the product. $$ A_{2 \times 3} \cdot B_{3 \times 2} $$

Step-by-Step Solution

Answer

The product is defined and the resulting matrix is \( 2 \times 2 \).

Step 1: Identify matrix dimensions

Matrix \( A \) has dimensions \( 2 \times 3 \), and matrix \( B \) has dimensions \( 3 \times 2 \). This means \( A \) has 2 rows and 3 columns, whereas \( B \) has 3 rows and 2 columns.

Step 2: Check matrix product compatibility

To determine if two matrices can be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, matrix \( A \) has 3 columns, and matrix \( B \) has 3 rows. Therefore, the matrices can be multiplied.

Step 3: Determine the dimensions of the resulting matrix

When two matrices can be multiplied, the resulting product matrix will have dimensions based on the number of rows from the first matrix and the number of columns from the second matrix. Thus, for \( A_{2 \times 3} \cdot B_{3 \times 2} \), the resulting matrix will have dimensions \( 2 \times 2 \).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in matrix multiplication. Each matrix is represented by rows and columns. For example, Matrix \( A \) has dimensions \( 2 \times 3 \), which means it has 2 rows and 3 columns. Matrix \( B \), on the other hand, has dimensions \( 3 \times 2 \), signifying 3 rows and 2 columns.

Key Points to Remember:

• Rows are the horizontal groups of numbers.
• Columns are the vertical groups of numbers.
• The dimension is written as rows first, then columns (e.g., \( m \times n \)).

Matrix Product Compatibility

Matrix product compatibility refers to whether two matrices can be multiplied. This is determined by comparing the dimensions of the matrices.

For two matrices to be compatible for multiplication:

• The number of columns in the first matrix must be the same as the number of rows in the second matrix.
• In our example, matrix \( A \) has 3 columns, and matrix \( B \) has 3 rows.

Since the columns of \( A \) match the rows of \( B \), these matrices are compatible for multiplication. This ensures that each element from the rows of \( A \) can pair with each element of the columns from \( B \) to form the product.

Resulting Matrix Dimensions

Once you've determined that two matrices can be multiplied, the next step is to identify the dimensions of the resulting matrix. The resulting matrix will have dimensions based on:

• The number of rows from the first matrix.
• The number of columns from the second matrix.

In the case of \( A_{2 \times 3} \cdot B_{3 \times 2} \), the resulting product matrix will have dimensions \( 2 \times 2 \). This is because:

• The number of rows from Matrix \( A \) is 2.
• The number of columns from Matrix \( B \) is also 2.

Hence, the product matrix will be \( 2 \times 2 \), containing four elements arranged accordingly.