Problem 14 - Algebra 2

Question

Determine whether each matrix product is defined. If so, state the dimensions of the product. \(A_{4 \times 3} \cdot B_{3 \times 2}\)

Step-by-Step Solution

Answer

The matrix product is defined and has dimensions 4 by 2.

Step 1: Identify the Matrices' Dimensions

Matrix A has dimensions 4 by 3, written as \(A_{4 \times 3}\). Matrix B has dimensions 3 by 2, written as \(B_{3 \times 2}\). In matrix multiplication, an \(m \times n\) matrix can be multiplied by an \(n \times p\) matrix. In this case, \(n\) is the number of columns in A, which is also the number of rows in B.

Step 2: Check Compatibility for Multiplication

To determine if two matrices can be multiplied, check if the number of columns in the first matrix equals the number of rows in the second matrix. Here, matrix A has 3 columns, and matrix B has 3 rows. Since these are equal, the matrices can be multiplied.

Step 3: Determine the Dimensions of the Product Matrix

When multiplying an \(m \times n\) matrix by an \(n \times p\) matrix, the resulting matrix will be an \(m \times p\) matrix. Since matrix A is 4 by 3 and matrix B is 3 by 2, the resulting matrix will have dimensions of 4 by 2 (\(4 \times 2\)).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in matrix multiplication. Each matrix can be described by its size, or dimensions, which are noted as "rows by columns." For example, if a matrix has 4 rows and 3 columns, it is written as \( 4 \times 3 \). These dimensions tell us not only the size of the matrix but also influence how matrices can interact with one another. To visualize this, you may think of each row as a horizontal line of numbers and each column as a vertical stack of numbers you can pluck information from.

• Rows: The horizontal lines of elements in a matrix.
• Columns: The vertical collections of elements.
• Notation: \( m \times n \) for a matrix with \( m \) rows and \( n \) columns.

When identifying a matrix for multiplication, it's vital to first write down and understand its dimensions, which play a role in subsequent steps.

Multiplicability of Matrices

Whether two matrices can be multiplied depends entirely on their dimensions. In matrix multiplication, we consider an operation between two matrices, say \( A \) and \( B \). Here, for valid multiplication, the number of columns in matrix \( A \) must equal the number of rows in matrix \( B \). This is because each element in the rows of \( A \) must align with each element in the columns of \( B \). If these conditions are not met, the matrices are deemed incompatible for multiplication.

• Check that the number of columns in the first matrix matches the rows of the second matrix.
• If dimensions described by \( A_{m \times n} \), then compatible with \( B_{n \times p} \).
• Incompatibility arises when the column-row requirement above is not satisfied.

This essential rule ensures that every element title in matrix \( A \) meets correctly with oppositional elements in matrix \( B \). Without meeting this criterion, the journey of multiplication cannot proceed.

Resulting Matrix Dimensions

Once it is confirmed that two matrices are indeed multiplicable, the next focus is on the dimensions of the resulting matrix. This resultant is determined by the number of rows in the first matrix and the number of columns in the second matrix. For instance, if matrix \( A \) is \( m \times n \) and matrix \( B \) is \( n \times p \), the resulting product will be an \( m \times p \) matrix. Visually, this can be imagined as each row of \( A \) coming together with each column of \( B \).

• The resulting number of rows: Taken from the first matrix.
• The resulting number of columns: Taken from the second matrix.
• Example: \( A_{4 \times 3} \) times \( B_{3 \times 2} \) results in a \( 4 \times 2 \) matrix.

Understanding how the dimensions of the resulting matrix are formed helps in anticipating the structure and organization of the new matrix after multiplication.