Problem 58 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a \(2 \times 4\) matrix and \(B\) is a matrix such that \(A B A\) is defined, then the size of \(B\) must be \(4 \times 2\).

Step-by-Step Solution

Answer

The statement is true. Given the dimensions of matrix \(A\) and the given condition that \(ABA\) is defined, we determined that matrix \(B\) must have dimensions \(4 \times 2\).

Step 1: Determine the dimensions of ABA

Matrix \(A\) is a \(2 \times 4\) matrix as given. Let \(B\) be a matrix of dimensions \(x \times y\). From the given statement, the product \(ABA\) is defined. So, let's compute the product step-by-step: First, we will compute the product \(AB\). For this to be possible, the number of columns in \(A\) must be equal to the number of rows in \(B\). Since \(A\) has \(4\) columns, \(B\) must have \(4\) rows. Thus, we know that \(x = 4\) and the equation becomes: $$AB = ( 2 × 4) × (4 × y)$$ The dimensions of the resulting matrix \(AB\) will be \(2 \times y\). Now we have to compute the product \(ABA\). As we know, the dimensions of \(AB\) are \(2 \times y\). Since we are multiplying \(AB\) on the right by matrix \(A\), for this product to be defined, the number of rows in \(A\) (a \(2 \times 4\) matrix) must be equal to the number of columns in \(AB\) which is \(y\). So, \(y = 2\). After finding the value of \(y\), we can rewrite the matrix \(B\) as a \(4 \times 2\) matrix.

Step 2: Verify the truth of the statement

Given the dimensions of matrix \(A\) and the given condition that \(ABA\) is defined, we determined that matrix \(B\) must have dimensions \(4 \times 2\). Therefore, the statement is true. If a matrix \(B\) exists such that \(ABA\) is defined, then the dimensions of matrix B must be \(4 \times 2\).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when it comes to matrix multiplication. The term 'dimension' of a matrix refers to its size, which is expressed by the number of rows and columns it has. For instance, a matrix with 2 rows and 4 columns is represented as a 2×4 matrix. This becomes particularly important during multiplication, as the number of columns in the first matrix must be equal to the number of rows in the second matrix for multiplication to be defined.

For example, consider that you have two matrices, A and B. If matrix A is a 2×4 matrix and you want to multiply it by matrix B, then matrix B must have 4 rows. The number of columns in matrix B will determine the size of the resulting matrix. If matrix B has 4 rows and 2 columns (4×2), then the multiplication AB is valid, and the resulting matrix will have dimensions of 2×2.

When we discuss the multiplication sequence of ABA, the dimensions are even more closely intertwined. In our exercise, since A is a 2×4 matrix and we know ABA is defined, B must logically be a 4×2 matrix. This ensures that AB can be computed and that BA, using the result of AB, can also be computed, since the matrix A can post-multiply a 2×y matrix if y equals 2.

Associative Property of Matrix Multiplication

The associative property is a fundamental principle in matrix algebra which allows us to regroup the parentheses in an expression without changing the result. According to the associative property of matrix multiplication, for any three matrices A, B, and C, where the products AB and BC are defined, the equality \( (AB)C = A(BC) \) holds. It's important to note that matrix multiplication is not commutative (i.e., \( AB \) does not necessarily equal \( BA \) ), but it is associative, which is pivotal in simplifying complex matrix expressions.

In the context of our exercise, this property assures us that we can compute ABA by first calculating AB and then multiplying the result by A, or alternatively by computing BA and then multiplying that result by A on the left. However, it's only possible to use the associative property if the dimensions of the matrices align properly to perform the multiplications. This constraint is precisely why we deduced that B must be a 4×2 matrix; only then the dimensions match for the multiplication to be well-defined.

Matrix Algebra

Matrix algebra involves operations including addition, subtraction, and multiplication of matrices, as well as the application of more complex operations such as finding the determinant or the inverse of a matrix. Underlying these operations are the rules dictated by matrix dimensions, as discussed before. In the case of multiplication, these rules are strict: only matrices with compatible dimensions can be multiplied.

When we work within the realm of matrix algebra, besides knowing the operations, understanding the properties of these operations--like the associative, distributive, and identity properties--can be pivotal. In our given exercise, recognizing and applying the dimension compatibility rule was key. Without a strong grasp of these foundational concepts, such as ensuring the inner dimensions match, one cannot move forward to more advanced operations in matrix algebra, nor correctly interpret the results of these operations.

In essence, matrix algebra is a systematic way of applying mathematical operations to matrices, and fluency in it requires both an understanding of the operations themselves and the rules that must be followed for these operations to be valid.