Problem 29 - Calculus for Biology and Medicine

Question

Suppose that \(A\) is a \(3 \times 4\) matrix, \(B\) is a \(1 \times 3\) matrix, \(C\) is a \(3 \times 1\) matrix, and \(D\) is a \(4 \times 3\) matrix. Which of the matrix multiplications that follow are defined? Whenever it is defined, state the size of the resulting matrix. (a) \(B D^{\prime}\) (b) \(D A\) (c) \(A C B\)

Step-by-Step Solution

Answer

(a) is defined as \(1 \times 4\); (b) is defined as \(4 \times 4\); (c) is not defined.

Step 1: Understand the Matrix Multiplication Rule

Before solving each part, remember that matrix multiplication is only defined if the number of columns in the first matrix equals the number of rows in the second matrix. The size of the resulting matrix will be the number of rows of the first matrix by the number of columns of the second matrix.

Step 2: Analyze Multiplication (a) \(B D^{\prime}\)

Matrix \(B\) is a \(1 \times 3\) matrix. The transpose of matrix \(D\), denoted as \(D^{\prime}\), changes \(D\) from a \(4 \times 3\) to a \(3 \times 4\) matrix. The multiplication \(B D'\) is defined because the number of columns in \(B\) (3) matches the number of rows in \(D'\) (3). The resulting matrix size is \(1 \times 4\).

Step 3: Analyze Multiplication (b) \(D A\)

Matrix \(D\) is a \(4 \times 3\) matrix, and matrix \(A\) is a \(3 \times 4\) matrix. The multiplication \(D A\) is defined because the number of columns in \(D\) (3) matches the number of rows in \(A\) (3). The resulting matrix size is \(4 \times 4\).

Step 4: Analyze Multiplication (c) \(A C B\)

First, consider multiplying matrices \(A\) and \(C\). Matrix \(A\) is \(3 \times 4\) and matrix \(C\) is \(3 \times 1\). This multiplication is not defined, as the number of columns in \(A\) (4) does not match the number of rows in \(C\) (3). As this multiplication is not defined, \(A C B\) is also not defined.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in matrix operations, particularly excelling in matrix multiplication. A matrix is essentially a rectangular array of numbers, and its dimensions tell us its structure.

The dimensions of a matrix are represented by two numbers: rows and columns. For example, a matrix is said to have dimensions \( m \times n \) if it has \( m \) rows and \( n \) columns. This is read as "m by n."

• A \(3 \times 4\) matrix has 3 rows and 4 columns.
• A \(1 \times 3\) matrix has 1 row and 3 columns.
• A \(4 \times 3\) matrix, like some in our problem, has 4 rows and 3 columns.

Understanding matrix dimensions helps determine if two matrices can be multiplied. This depends on the inner dimensions matching: the number of columns in the first matrix must equal the number of rows in the second matrix.

Once multiplication is confirmed to be possible, the dimensions of the resulting matrix come from the outer dimensions:

• If multiplying a \(3 \times 4\) matrix by a \(4 \times 2\) matrix, the result will be a \(3 \times 2\) matrix.

Matrix Transpose

Matrix transposition is an operation that flips a matrix over its diagonal. This means that the rows become columns and the columns become rows.

The transpose of a matrix is denoted with a "prime" symbol, often referred to as a "dash" or "tick." For example, the transpose of matrix \( D \) is written as \( D^\prime \).

• For a \(4 \times 3\) matrix like \( D \), its transpose \( D^\prime \) would be \(3 \times 4\). This is because each of the 4 columns in the original becomes a row in the transpose, and each of the 3 rows becomes a column.

Understanding matrix transpose is particularly important when dealing with scenarios where multiplication isn't directly possible. By transposing, you can switch the matrix dimensions to fit multiplication requirements, as was done in part \(a\) of our given task.

In general, transposing a matrix does not change the data, just its orientation. It is as if you rotated the matrix and adjusted how you view it.

Matrix Size Determination

Matrix size determination refers to establishing the dimensions of the resulting matrix after performing multiplication. This not only requires checking compatibility but also mastering the technique to predict result sizes properly.

To determine the size of the product of two matrices, ensure compatibility first. This is done by confirming that the columns in the first matrix equal the rows in the second. If they match, you can then determine the size of the resulting matrix using the outer dimensions.

• For example, multiplying a \(1 \times 3\) matrix by a \(3 \times 4\) matrix results in a \(1 \times 4\) matrix. Here, the inner dimensions (the 3s) confirm multiplication compatibility, and the outer dimensions tell the result's size.

However, if the matrices aren't compatible for multiplication, no result size determination occurs because the operation is not feasible. For example, if trying to multiply matrices where the number of columns in the first does not equal the number of rows in the second (like \(A\) and \(C\) in our task), then multiplication is defined as not possible, and no matrix results.