Problem 105 - Precalculus

Question

If \(A_{m \times n} B_{n \times p}\) is defined, explain why \(\left(A_{m \times n} B_{n \times p}\right)^{2}\) is not defined for \(m \neq p\)

Step-by-Step Solution

Answer

\((AB)^2\) is not defined for \(m \neq p\) because the resulting matrices from \(AB\) cannot be multiplied due to incompatible dimensions.

Step 1: Determine if Matrix Multiplication Exists

The product of two matrices, \( A_{m \times n} \) and \( B_{n \times p} \), is defined if the number of columns in the first matrix \( A \) matches the number of rows in the second matrix \( B \). This condition is satisfied as \( A \) has \( n \) columns and \( B \) has \( n \) rows. Thus, the product \( AB \) results in a matrix of dimension \( m \times p \).

Step 2: Check Dimensions for Squaring the Product

To find \( (AB)^2 \), you must multiply \( AB_{m \times p} \) by itself. For matrix multiplication \( AB \cdot AB \) to exist, the number of columns in the first \( AB_{m \times p} \) must equal the number of rows in the second \( AB_{m \times p} \). Hence, \( p \) must equal \( m \).

Step 3: Understand Why the Product is Undefined

If \( m eq p \), then the multiplication \( (AB)_{m \times p} \times (AB)_{m \times p} \) is not defined because the condition for matrix multiplication \( (p = m) \) is not satisfied.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when working with matrices. Each matrix is defined by its dimensions, represented as rows by columns, denoted as \(m \times n\). Here, \(m\) indicates the number of rows, while \(n\) indicates the number of columns.
For example:

• A matrix \(A\) with 3 rows and 4 columns is represented as \(3 \times 4\).
• Another matrix \(B\) with 5 rows and 2 columns is \(5 \times 2\).

Recognizing matrix dimensions helps in deciding whether operations such as addition, subtraction, or multiplication are possible. It's important to always place the row dimension before the column dimension. This ordering is vital in ensuring the correct execution of matrix operations.

Matrix Product Conditions

Matrix multiplication is an operation that combines two matrices to produce a new matrix. However, it's not always possible to multiply any two matrices. Certain conditions must be satisfied for the product to be defined. Specifically, if you have matrix \(A_{m \times n}\) and matrix \(B_{n \times p}\), the multiplication \(AB\) is only possible when the number of columns in \(A\) equals the number of rows in \(B\).
This condition ensures the compatibility between the matrices:

• The resulting matrix from the multiplication will have dimensions \(m \times p\).
• Each element of the resulting matrix is calculated as the dot product of the respective row from matrix \(A\) and column from matrix \(B\).

When these product conditions are not satisfied, such as when the number of columns in \(A\) does not match the number of rows in \(B\), the matrix multiplication operation cannot be performed.

Squaring Matrices

Squaring a matrix involves multiplying the matrix by itself. However, to square a matrix, it must first be eligible for multiplication with itself, meaning that it should possess the correct dimensions. If we're attempting to square the product matrix \((AB)\), which has dimensions \(m \times p\), certain conditions must apply.
The dimensions dictate that for \((AB)^2\) to be defined, the matrices should satisfy the rule: the number of columns of the first \((AB)_{m \times p}\) should match the number of rows of the second \((AB)_{p \times m}\). Therefore, \(p\) must equal \(m\).
In scenarios where \(m eq p\), the multiplication \((AB) \cdot (AB)\) cannot be performed. This is because the matrices involved do not meet the required condition of having compatible dimensions for multiplication. Therefore, the square of a product matrix is undefined unless these conditions are met. Knowing when and how matrices can be squared or multiplied is crucial for correctly solving more complex problems in linear algebra.