Problem 54 - Precalculus: Mathematical 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

Both matrices must be square matrices of the same dimension.

Step 1: Understand Matrix Multiplication

Matrix multiplication is only defined when the number of columns in the first matrix matches the number of rows in the second matrix. That is, for two matrices \( A \) (with dimensions \( m \times n \)) and \( B \) (with dimensions \( p \times q \)), the product \( AB \) is defined only if \( n = p \).

Step 2: Define AB

For the product \( AB \) to be defined, the number of columns in \( A \) must equal the number of rows in \( B \). If \( A \) is an \( m \times n \) matrix, then \( B \) must be an \( n \times q \) matrix.

Step 3: Define BA

For the product \( BA \) to be defined, the number of columns in \( B \) must equal the number of rows in \( A \). If \( B \) is an \( n \times q \) matrix, then \( A \) must be a \( q \times m \) matrix.

Step 4: Find Common Condition

Since both \( AB \) and \( BA \) are defined, \( n = p \) and \( q = m \) must hold true simultaneously. This implies both matrices \( A \) and \( B \) must be square matrices with dimensions \( n \times n \).

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial when it comes to performing operations like matrix multiplication. They are represented by two numbers: the number of rows and the number of columns. Whenever you see dimensions in the context of matrices, you'll often see something like "m x n", where "m" refers to the number of rows and "n" refers to the number of columns.

For instance, in order to multiply two matrices together, the number of columns in the first matrix must match the number of rows in the second matrix. This rule ensures that each element from a row in the first matrix can multiply with the corresponding element from a column in the second matrix, allowing for the matrix product to be calculated.

Therefore, a matrix with dimensions "3 x 4" has 3 rows and 4 columns, and if you want to multiply this matrix with another, the second matrix must start with 4 rows. This means its dimension must be something like "4 x k" where "k" is any number of columns in the second matrix. Understanding matrix dimensions is key to knowing when matrix operations can be performed.

Square Matrices

Square matrices are matrices where the number of rows and columns are equal. For example, a "3 x 3" matrix has three rows and three columns, making it square. Square matrices are significant in many mathematical applications.

One important application is in matrix multiplication. When both matrices are square, the rules for multiplying matrices become simpler, because both products \( AB \) and \( BA \) can be defined as long as each matrix is of the dimension \( n \times n \).

Square matrices also have unique properties that are not present in other matrices, such as the possibility of calculating the determinant and checking for invertibility.

• Determinant: A unique number that can be computed from the elements of a square matrix.
• Invertibility: A square matrix can sometimes have an inverse, which is a matrix that, when multiplied with the original, results in the identity matrix. Not all square matrices have inverses, but only those with a non-zero determinant.

Understanding the concept of square matrices can therefore unlock a wide range of mathematical techniques and applications.

Matrix Product

Matrix product, also known as matrix multiplication, is a fundamental operation in linear algebra. It involves combining two matrices to produce a third matrix. For two matrices to be multiplied together, specific conditions regarding their dimensions must be met.

As mentioned earlier, if you have a matrix \( A \) with dimensions \( m \times n \), and you want to multiply it by matrix \( B \), matrix \( B \) must have dimensions that start with \( n \), such as \( n \times p \). This allows each element of each row in matrix \( A \) to interact with each element of each column in matrix \( B \), creating a new set of numbers that form the matrix product.

Moreover, the resulting matrix from their multiplication will have dimensions made up from the outer numbers of the matrices involved. So continuing our example, matrix \( C = AB \) will have the dimensions \( m \times p \).

Here's what happens step-by-step when multiplying matrices:

• Multiply corresponding elements and sum them up.
• Repeat this row by column multiplication for the entire row of first matrix and the entire column of the second matrix.

This step-by-step interaction between rows and columns results in a matrix that summarizes the relationships represented by the original matrices. Understanding these principles is vital for effectively working with matrix operations.