Problem 61 - Precalculus: Mathematics for Calculus

Question

What must be true about the dimensions of the matrices \(A\) and \(B\) if both products \(A B\) and \(B A\) are defined?

Step-by-Step Solution

Answer

For \( AB \) and \( BA \) to be defined, A must be \( m \times n \) and B must be \( n \times m \).

Step 1: Understand Matrix Multiplication

For matrices, the multiplication is defined when the number of columns in the first matrix is equal to the number of rows in the second matrix. This is the basic rule for the product of two matrices to be defined.

Step 2: Define Dimensions of Matrix A and B

Let's denote the dimensions of matrix A as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. Similarly, denote the dimensions of matrix B as \( p \times q \), where \( p \) is the number of rows and \( q \) is the number of columns.

Step 3: Conditions for AB to be Defined

The product \( AB \) will be defined if the number of columns of A (which is \( n \)) is equal to the number of rows of B (which is \( p \)). Therefore, we require \( n = p \).

Step 4: Conditions for BA to be Defined

The product \( BA \) will be defined if the number of columns of B (which is \( q \)) is equal to the number of rows of A (which is \( m \)). Therefore, we require \( q = m \).

Step 5: Combine All Conditions

For both products \( AB \) and \( BA \) to be defined, the dimensions must satisfy two conditions: \( n = p \) and \( q = m \). This means the number of columns in A must match the number of rows in B, and the number of columns in B must match the number of rows in A.

Key Concepts

Matrix Dimensions

Understanding the dimensions of a matrix is crucial when working with matrix multiplication. A matrix is essentially a rectangular array of numbers arranged in rows and columns, and its dimensions are typically denoted as \( m \times n \). In this notation, \( m \) refers to the number of rows, and \( n \) indicates the number of columns. For instance, a 3x4 matrix has 3 rows and 4 columns. Knowing the dimensions is important as it helps determine whether certain operations, like matrix multiplication, are possible.

• Rows (\( m \)): The horizontal arrangement in the matrix.
• Columns (\( n \)): The vertical arrangement in the matrix.

To check if two matrices can be multiplied, one needs to look at the dimensions of each matrix involved. Each matrix has its specific row and column counts, which together define its size and mathematical properties.

Matrix Product

The matrix product, also known as matrix multiplication, is a binary operation that takes two matrices and produces another matrix. However, unlike numerical multiplication, not all matrices can be multiplied by each other. For the matrix product to be defined, specific conditions based on matrix dimensions must be fulfilled. When multiplying two matrices, such as matrix \( A \) with dimensions \( m \times n \) and matrix \( B \) with dimensions \( p \times q \), the matrix product \( AB \) is only possible if the number of columns of matrix \( A \) (which is \( n \)) matches the number of rows of matrix \( B \) (which is \( p \)). The resulting matrix \( C \) will then have dimensions \( m \times q \).

• The first condition is often phrased as "columns of the first equals rows of the second".
• The resulting matrix takes the row count of matrix \( A \) and the column count of matrix \( B \).

This multiplication is crucial in many applications, such as transforming geometrical data, solving linear equations, and computer graphics.

Multiplication Conditions

For multiplication to occur between two matrices, specific conditions regarding their dimensions must be met. This ensures that the matrix operations are mathematically valid and computable. Let’s break down these conditions for clarity. For matrices \( A \) and \( B \) to be multiplied in both directions, thus yielding \( AB \) and \( BA \), they must satisfy the following dimensional constraints:

• For \( AB \): The number of columns in \( A \), denoted by \( n \), must be equal to the number of rows in \( B \), denoted by \( p \). This means \( n = p \).
• For \( BA \): The number of columns in \( B \), denoted by \( q \), must match the rows in \( A \), denoted by \( m \). So, \( q = m \).

If both of these conditions are satisfied, both products \( AB \) and \( BA \) are definable. Meeting these conditions allows both products to exist, which can be pivotal in complex mathematical computations and represents a fundamental property in linear algebra.