Problem 16 - Calculus for Biology and Medicine

Question

Suppose \(A\) is a \(2 \times 2\) matrix. Find conditions on the entries of \(A\) such that $$ A+A^{\prime}=\mathbf{0} $$

Step-by-Step Solution

Answer

The matrix \( A \) must be of the form \( \begin{pmatrix} 0 & b \\ -b & 0 \end{pmatrix} \).

Step 1: Understand the Matrix Notation

Matrix \( A \) is a \( 2\times 2 \) matrix, meaning it has two rows and two columns. We can represent it as \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) where \( a, b, c, \) and \( d \) are real numbers.

Step 2: Define the Transpose of the Matrix

The transpose of a matrix, denoted \( A' \) or \( A^T \), is obtained by flipping the matrix over its diagonal. Therefore, the transpose of \( A \), \( A' = \begin{pmatrix} a & c \ b & d \end{pmatrix}\).

Step 3: Set up the Equation

The given condition is \( A + A' = \mathbf{0} \), where \( \mathbf{0} \) is the \( 2\times 2 \) zero matrix, \( \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \). Substituting \( A \) and \( A' \), the equation becomes \( \begin{pmatrix} a & b \ c & d \end{pmatrix} + \begin{pmatrix} a & c \ b & d \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \).

Step 4: Simplify the Matrix Addition

Add the corresponding elements of \( A \) and \( A' \): \[ \begin{pmatrix} a + a & b + c \ c + b & d + d \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \].

Step 5: Solve for Matrix Entry Conditions

Set each entry of the resulting matrix to zero:\( 2a = 0 \), \( b + c = 0 \), \( b + c = 0 \), and \( 2d = 0 \). Solving these, we find \( a = 0 \), \( b = -c \), and \( d = 0 \).

Step 6: Conclusion

The conditions on the entries of \( A \), for \( A + A' = \mathbf{0} \) to hold true, are that \( a = 0 \), \( d = 0 \), and \( b = -c \). Thus, \( A \) must be of the form \( \begin{pmatrix} 0 & b \ -b & 0 \end{pmatrix} \).

Key Concepts

Matrix Transpose

One of the fundamental operations in matrix algebra is taking the transpose of a matrix. To transpose a matrix, you swap its rows and columns. For example, if you have a matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), its transpose, denoted \( A^T \) or \( A' \), would be \( \begin{pmatrix} a & c \ b & d \end{pmatrix} \).
Here are some key points about matrix transpose:

• The transpose of a transpose returns the original matrix. That is, \((A^T)^T = A\).
• For any matrix, the element at the \((i, j)\) position of \(A^T\) is at the \((j, i)\) position of \(A\).
• The transpose of a square matrix (same number of rows and columns) will still be a square matrix.
• Transposing does not change the determinant of a matrix, i.e., \( \det(A) = \det(A^T) \).

Understanding transposition is crucial, especially when exploring concepts like symmetry and operations leading to zero matrices. It allows you to explore the properties of matrices and their behaviors in different operations.

Symmetric Matrix

A symmetric matrix is a special type of square matrix where the elements are symmetric about the main diagonal. In simpler terms, each element of the matrix is mirrored across the diagonal. If you have a matrix \( A \), it is symmetric if and only if its transpose equals itself, i.e., \( A = A^T \).
For example, a symmetric matrix looks like this: \( \begin{pmatrix} x & y \ y & z \end{pmatrix} \). Here, the element \( y \) is repeated symmetrically across the diagonal.Some important properties of symmetric matrices include:

• Symmetric matrices always have real eigenvalues.
• They can be decomposed into a diagonal form using an orthogonal transformation.
• Symmetric matrices are particularly useful in optimization problems and systems of linear equations, where their properties simplify calculations and solutions.

In the exercise, the given condition, when followed, leads to a form that is anti-symmetric, a concept closely related to symmetric matrices. Specifically, the off-diagonal elements are negatives of each other, which alludes to the matrix having unique properties vis-à-vis symmetry.

Zero Matrix

A zero matrix, also known as the null matrix, is a matrix where all of its elements are zero. It is often denoted as \( \mathbf{0} \). For a matrix to be a zero matrix, irrespective of its size, every entry must be zero: \( \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \).
Zero matrices serve as the additive identity in matrix operations, which means any matrix plus the zero matrix will return the original matrix. For any matrix \( A \), \( A + \mathbf{0} = A \).Key points regarding zero matrices include:

• A zero matrix can have any dimensions, from a \(1 \times 1\) to an \(n \times n\).
• When you multiply any matrix by a zero matrix, regardless of the order, the result is a zero matrix.
• In equations involving matrices, zero matrices often simplify to provide critical conditions or constraints, as observed in expressions like \( A + A' = \mathbf{0} \).

Understanding zero matrices is essential because they act as placeholders and help define identities in matrix algebra. In the exercise, the resulting form of the matrix after operations shows how zero matrices provide structures and conditions for algebraic operations to hold true.