Problem 50 - Algebra and Trigonometry

Question

When Are Both Products Defined? 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 products are defined if \(A\) is \(m \times n\) and \(B\) is \(n \times m\).

Step 1: Understand Matrix Multiplication

To multiply two matrices, the number of columns of the first matrix must equal the number of rows of the second matrix. Thus, to multiply matrix \(A\) (with dimensions \(m \times n\)) by matrix \(B\) (with dimensions \(p \times q\)), we require that \(n = p\).

Step 2: Check Conditions for AB

For the product \(AB\) to be defined, the number of columns of \(A\) must equal the number of rows of \(B\). If \(A\) is \(m \times n\), and \(B\) is \(p \times q\), then we need \(n = p\) for \(AB\) to be defined.

Step 3: Check Conditions for BA

For the product \(BA\) to be defined, the number of columns of \(B\) must equal the number of rows of \(A\). Given \(A\) is \(m \times n\), and \(B\) is \(p \times q\), we need \(q = m\) for \(BA\) to be defined.

Step 4: Combine Conditions

Both products \(AB\) and \(BA\) will be defined only if \(n = p\) and \(q = m\). Thus, \(A\) must be \(m \times n\) and \(B\) must be \(n \times m\).

Key Concepts

Matrix Dimensions

When we talk about matrices, it's crucial to understand their dimensions. A matrix's dimensions are given in a format of "rows by columns." For example, if a matrix has 3 rows and 4 columns, its dimensions are represented as \(3 \times 4\). Understanding matrix dimensions is essential because it determines whether certain operations, like multiplication, are possible.
The dimensions give you a roadmap for how to interact with the matrix during mathematical operations. When multiplying matrices, you need to be particularly mindful of these dimensions. The number of columns in the first matrix must match the number of rows in the second matrix. Hence, even if two matrices have compatible individual dimensions, matrix multiplication isn’t automatically defined unless these conditions are met. With these rules in mind, mastering matrix dimensions can make complex operations like matrix products much more manageable.

Matrix Products

Matrix products, or the resulting matrices when you multiply two matrices, can be confusing at first. Once you understand the basic rules, they're easier to grasp. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. If matrix \(A\) is \(m \times n\) and matrix \(B\) is \(p \times q\), you can only multiply them if \(n = p\). The resulting matrix will have dimensions \(m \times q\).
To determine if both product orders \(AB\) and \(BA\) are defined, as is the case in the original exercise, the conditions \(n = p\) for \(AB\) and \(q = m\) for \(BA\) must both be met. In other words, for \(AB\) and \(BA\) to both exist, matrix \(A\) must be \(m \times n\) and matrix \(B\) must be \(n \times m\). When these conditions are met, you've ensured the products are defined.

Algebra

Matrix multiplication is a fascinating extension of algebra, where numbers are replaced by matrices. It follows a specific set of rules that can be seen as abstract algebraic properties applied to matrices. The associative property holds, meaning that matrix multiplication is associative, i.e., \((AB)C = A(BC)\). However, it's important to note that matrix multiplication is not commutative. This means that generally, \(AB eq BA\).
Understanding these algebraic properties helps you navigate the rules that dictate matrix behavior. Knowing them will not only aid in understanding multiplication but will also form the foundation for more advanced algebraic studies involving matrices, such as determinants and inverses. Remembering that order and dimension compatibility affect the overall feasibility of multiplication goes a long way in mastering the algebra of matrices.