Problem 20 - Calculus for Biology and Medicine

Question

Suppose that \(A\) is an \(m \times k\) matrix and \(B\) is a \(k \times n\) matrix. Show that $$ (A B)^{\prime}=B^{\prime} A^{\prime} $$

Step-by-Step Solution

Answer

We proved \((AB)' = B'A'\) using the transpose property of matrix products.

Step 1: Understand the Transpose of a Product

We need to prove the relationship \((A B)' = B' A'\). Recall that the transpose of a product follows the rule that \((XY)' = Y'X'\) when the matrices are compatible for multiplication. Here, \(A\) is an \(m \times k\) matrix and \(B\) is a \(k \times n\) matrix, making \(AB\) an \(m \times n\) matrix.

Step 2: Apply the Transpose Property to Matrices A and B

According to the property of transpose, for the matrix product \(AB\), the transpose of the product is obtained by reversing the order of multiplication and transposing the individual matrices. Therefore, we write: \((AB)' = B'A'\).

Step 3: Conclude the Solution

By carefully applying the transpose property of matrix products, we have shown that \((AB)' = B'A'\), proving the original statement. This demonstration confirms the mathematical property of transposes in matrix algebra.

Key Concepts

Matrix Transpose

In matrix algebra, the concept of a transpose is fundamental. It involves transforming a matrix by flipping it over its diagonal. This process interchanges the row and column indices of the matrix. For a given matrix \( C \), denoted as \( C^T \), every element \( c_{ij} \) is repositioned as \( c_{ji} \). In simple terms:

• Rows become columns
• Columns become rows

For example, if \( C \) is a 2x3 matrix, its transpose \( C^T \) will be a 3x2 matrix. This method not only changes the position of elements but also the dimensions of the matrix.

An essential rule to remember is that for transpose operations in matrix products, the order of multiplication reverses. So, \((XY)^T = Y^TX^T\) holds, and this rule is particularly useful in solving and simplifying algebraic expressions involving matrices.

Matrix Multiplication

Matrix multiplication is a bit different from multiplying regular numbers. It involves combining two matrices to produce another matrix. However, not every pair of matrices can be multiplied. For two matrices \(A\) and \(B\) to be multiplied, the number of columns in \(A\) must match the number of rows in \(B\).

When multiplying matrices:

• The resultant element at position \(i, j\) in the new matrix results from the dot product between the \(i^{th}\) row of the first matrix and the \(j^{th}\) column of the second matrix.
• The dimensions of the resulting matrix are determined by taking the rows from the first matrix and columns from the second, i.e., for an \(m \times k\) matrix \(A\) and a \(k \times n\) matrix \(B\), the product \(AB\) will be an \(m \times n\) matrix.

Matrix multiplication is associative [\((AB)C = A(BC)\)], but not necessarily commutative [\(AB eq BA\)], unless under specific circumstances when both matrices are square and satisfy certain conditions.

Matrices

Matrices are a powerful tool in mathematics, particularly useful for representing linear transformations, solving systems of equations, and much more. A matrix is essentially a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Each item in a matrix is called an element.

There are several types of matrices, including:

• Square Matrix: The same number of rows and columns, such as a 3x3 matrix.
• Row Matrix: A matrix with only one row, e.g., 1x3.
• Column Matrix: A matrix with only one column, e.g., 3x1.
• Zero Matrix: A matrix where all elements are zero.
• Identity Matrix: A square matrix with ones on the diagonal and zeros elsewhere.

Understanding matrices is crucial because they serve as the backbone for more complex mathematical concepts, such as linear transformations, and are used extensively in fields like computer science, physics, and engineering.