Problem 19 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The transpose of a scalar-multiplied matrix \( kA \) is equal to \( k \) times the transpose of \( A \).

Step 1: Understand Matrix Scalar Multiplication

The scalar multiplication of a matrix involves multiplying every entry of the matrix by a scalar constant. If \( A \) is an \( m \times n \) matrix, then \( kA \) represents a matrix where every element of \( A \) is multiplied by \( k \). For example, if \( A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \), then \( kA = \begin{pmatrix} ka_{11} & ka_{12} \ ka_{21} & ka_{22} \end{pmatrix} \).

Step 2: Understand Matrix Transposition

Transposing a matrix involves swapping its rows and columns. For an \( m \times n \) matrix \( A \), its transpose \( A^{\prime} \) is an \( n \times m \) matrix. Elements are rearranged such that the element at position \((i, j)\) in \( A \) is at position \((j, i)\) in \( A^{\prime} \).

Step 3: Transpose of Scalar-Multiplied Matrix

When we transpose the scalar-multiplied matrix \( (kA) \), we apply the transposition to \( kA \). Since scalar multiplication distributes over matrix transposition, \( (kA)^{\prime} = (kA_{ij})^{\prime} = (ka_{ij})^{\prime} \), resulting in \( (ka_{ji}) \). Also, transposition affects only the position of elements, not the scalar factor.

Step 4: Proof of the Property

The transpose of \( kA \), denoted \((kA)^{\prime}\), results in the matrix \( kA^{\prime} \) where its elements are \( k \times a_{ji} \). This shows that \((kA)^{\prime} = k(A^{\prime})\), representing the fact that the scalar factor \( k \) can be factored out of the transpose operation, and scalar multiplication is preserved after transposition.

Key Concepts

Scalar Multiplication

Scalar multiplication is a basic operation in linear algebra, where a real number, known as a scalar, is used to multiply each element within a matrix. This operation uniformly scales the matrix elements by the scalar. Take, for example, a matrix \( A \) of size \( m \times n \). If we multiply \( A \) by a scalar \( k \), this results in a new matrix, \( kA \), where each element of \( A \) is multiplied by \( k \).
If \( A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \), then \( kA = \begin{pmatrix} ka_{11} & ka_{12} \ ka_{21} & ka_{22} \end{pmatrix} \).
Simply put, every single entry in the original matrix becomes \( k \) times its original amount. This operation is crucial when manipulating matrices in larger linear algebra computations, like solving systems of equations.

Matrix Algebra

Matrix algebra refers to a set of operations you can perform on matrices, including addition, subtraction, multiplication, and transposition. These operations are the building blocks of more complex linear algebra concepts.
In matrix algebra, addition and subtraction require matrices to be of identical dimensions. Multiplication is slightly more complex. Two matrices can be multiplied only if the number of columns in the first matrix matches the number of rows in the second.

• Given matrices \( A \) and \( B \), the product matrix \( C = AB \) will have dimensions corresponding to the rows of \( A \) and the columns of \( B \).
• The transposition of a matrix, represented as \( A^{\prime} \), involves swapping its rows and columns.

Matrix algebra empowers us to efficiently handle systems, transformations, and various problems in applied mathematics and engineering.

Linear Algebra Proofs

In linear algebra, proofs are a logical demonstration of the truth of a particular statement or proposition. These proofs often involve properties of matrices and operations performed on them. A common task is to prove identities, like the commutativity of operations or the interaction between matrix manipulation techniques such as scalar multiplication and transposition.
For example, to prove \((k A)^{\prime} = k A^{\prime}\), we rely on our understanding that transposition rearranges matrix elements, swapping positions \((i, j)\) with \((j, i)\) without affecting a scalar.

• The property \((kA)^{\prime} = k(A^{\prime})\) shows the scalar \( k \) does not change position but multiplies all elements of the transposed matrix \( A^{\prime} \).

Such proofs are fundamental for validating the properties and behaviors of linear algebra expressions, and they form a critical part of understanding and working with matrices in theoretical and practical applications.