Problem 19 - Algebra 2

Question

Determine whether each matrix product is defined. If so, state the dimensions of the product. \(A_{3 \times 1} \cdot B_{1 \times 5}\)

Step-by-Step Solution

Answer

The product is defined, with dimensions \(3 \times 5\).

Step 1: Identifying Matrices Dimensions

Matrix \(A\) is given as \(3 \times 1\) which means it has 3 rows and 1 column. Matrix \(B\) is given as \(1 \times 5\) which means it has 1 row and 5 columns.

Step 2: Checking Multiplicability

According to matrix multiplication rules, two matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix. Here, Matrix \(A\) has 1 column and Matrix \(B\) has 1 row, so they can be multiplied.

Step 3: Determining the Product Dimensions

Once we establish that the matrices can be multiplied, the dimensions of the resulting product matrix are determined by the number of rows in the first matrix and the number of columns in the second matrix. Thus, the product of \(A_{3 \times 1} \times B_{1 \times 5}\) will have dimensions \(3 \times 5\).

Key Concepts

Matrix Dimensions

When working with matrices, understanding their dimensions is crucial. The dimensions of a matrix are defined by the number of rows and columns it possesses. We denote these dimensions using a notation such as "3 \(\times\) 1", where the first number represents the number of rows, and the second number denotes the number of columns.

To illustrate, let's consider two matrices, \(A\) and \(B\). Matrix \(A\) is described as \(3 \times 1\), indicating it has 3 rows and a single column. Matrix \(B\), on the other hand, is \(1 \times 5\), having 1 row and 5 columns. Recognizing matrix dimensions is essential, particularly when it comes to operations like multiplication, and it sets the groundwork for understanding further matrix operations.

Matrix Product

The result of multiplying two matrices is known as the matrix product. To perform matrix multiplication, we adhere to a specific set of criteria regarding matrix dimensions. When two matrices are multiplied, the number of columns in the first matrix must match the number of rows in the second matrix. This rule is the cornerstone of determining when a matrix product is defined.

If these conditions are satisfied, the dimensions of the matrix product are derived from the remaining dimensions: the resulting matrix will have the number of rows of the first matrix and the number of columns of the second matrix.

• Matrix \(A\) has 3 rows and 1 column.
• Matrix \(B\) has 1 row and 5 columns.

Since the number of columns in \(A\) (1) equals the number of rows in \(B\) (1), the matrix product is defined. Hence, the resulting matrix, when \(A\) is multiplied by \(B\), will have dimensions \(3 \times 5\).

Matrix Multiplication Rules

Matrix multiplication follows certain rules that must be carefully observed to carry out multiplication successfully. A common rule is that the number of columns in the first matrix must equal the number of rows in the second matrix for the operation to be valid. This ensures that each element of the rows in the first matrix aligns properly with the columns in the second matrix for multiplication.

These elements are then summed to form each element of the resulting matrix. For example, the element in the first row and first column of the product is obtained by multiplying each element of the first row of the first matrix by the corresponding element of the first column of the second matrix and summing the results.

• Identify matrix dimensions.
• Ensure compatibility for multiplication.
• Determine resulting product dimensions.

Understanding and adhering to these rules not only clarifies when a matrix product can be computed but also prevents mistakes in the multiplication process.