Problem 95 - Precalculus

Question

Determine whether the statements are true or false. \(A B\) is defined only if the number of columns in \(A\) equals the number of rows in \(B\)

Step-by-Step Solution

Answer

True, the statement is correct as it describes a necessary condition for matrix multiplication.

Step 1: Understanding Matrix Multiplication

Matrix multiplication involves two matrices, say matrix \( A \) of size \( m \times n \) and matrix \( B \) of size \( p \times q \). For the multiplication \( AB \) to be defined, the number of columns in \( A \) (\( n \)) must match the number of rows in \( B \) (\( p \)).

Step 2: Analyzing the Given Statement

The statement given is that \( AB \) is defined only if the number of columns in \( A \) equals the number of rows in \( B \). Based on the requirement for matrix multiplication, this condition is necessary and sufficient for the product to be defined.

Step 3: Conclusion about the Statement

Since the condition mentioned in the problem aligns perfectly with the matrix multiplication requirement, the statement is true. The number of columns in \( A \) must indeed match the number of rows in \( B \) to define \( AB \).

Key Concepts

Matrices

Matrices are fundamental elements in mathematics, especially in linear algebra. They are rectangular arrays of numbers, symbols, or expressions, organized in rows and columns. Typically, a matrix is denoted by a capital letter, such as matrix \( A \) or matrix \( B \). Each value within a matrix is called an element or entry. These elements are defined by two indices: one for the row and one for the column.

For example, a matrix \( A \) with dimensions \( m \times n \) has \( m \) rows and \( n \) columns, which means it contains \( m imes n \) elements in total. The beauty of matrices lies in their capability to represent complex systems and transformations, making them versatile tools across various scientific, engineering, and economic fields.

• Representation: matrices are depicted with brackets or parentheses.
• Elements: each entry within the matrix is identified by its row and column position, such as \( a_{ij} \) for the element at row \( i \) and column \( j \).

Matrices provide a structured way to solve systems of linear equations, perform transformations, and conduct operations like addition, subtraction, and multiplication, among others.

Matrix Dimensions

Matrix dimensions play a crucial role in understanding and performing matrix operations. The size of a matrix is referred to by its dimensions, commonly noted as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. Knowing these dimensions is essential for many matrix operations, especially when considering whether two matrices can be multiplied or not.

For a matrix \( A \) of dimensions \( 3 \times 2 \), it would have 3 rows and 2 columns. Meanwhile, a matrix \( B \) of dimensions \( p \times q \) allows us to determine if multiplication like \( AB \) or \( BA \) is possible. This is where the concept of matrix compatibility comes into play.

• Multiplication requirement: for matrices \( A \) and \( B \) to be multiplied, the number of columns in \( A \) must equal the number of rows in \( B \).
• Resultant dimension: if matrices \( A \) and \( B \) can be multiplied, the result will have dimensions \( m \times q \).

Understanding matrix dimensions simplifies the process of solving mathematical problems involving complex systems and optimizations.

Matrix Product

Matrix product is a specific operation involving two matrices, resulting in a new matrix. This operation, also known as matrix multiplication, requires precision and a clear understanding of matrix dimensions to determine if the product is possible. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second. When matched correctly, the result is a new matrix whose size is determined by the number of rows in the first matrix and the number of columns in the second.

Consider matrices \( A \) and \( B \). If \( A \) has dimensions \( m \times n \) and \( B \) has dimensions \( n \times p \), the product \( AB \) will create a new matrix of dimensions \( m \times p \). Each element in the resultant matrix is calculated by taking the corresponding row from the first matrix and the column from the second, multiplying the respective elements, and summing them up.

• Defined product: \( AB \) is valid if the number of columns in \( A \) matches the number of rows in \( B \).
• Element calculation: each element in the resulting matrix, say \( c_{ij} \), is found by summing the products of corresponding elements from row \( i \) of \( A \) and column \( j \) of \( B \).

Mastering matrix multiplication is essential for those working with systems of equations, transformations, and various mathematical models.