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

Matrix \(A\) is symmetric if \(b = c\).

Step 1: Define the Matrix A

Suppose the matrix \(A\) is given by \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\). Knowing this, we can understand that the elements of the matrix are \(a, b, c,\) and \(d\).

Step 2: Compute the Transpose of A

The transpose of a matrix \(A\), denoted \(A^{\prime}\) or \(A^T\), is obtained by interchanging its rows and columns. Therefore, for the matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the transpose is \(A^{\prime} = \begin{pmatrix} a & c \ b & d \end{pmatrix}\).

Step 3: Set Up the Equation A - A' = 0

We know from the problem statement that \(A - A^{\prime} = \mathbf{0}\), where \(\mathbf{0}\) denotes the zero matrix. This gives us the equation: \(\begin{pmatrix} a & b \ c & d \end{pmatrix} - \begin{pmatrix} a & c \ b & d \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}\).

Step 4: Solve for Individual Matrix Elements

Subtract corresponding elements from both matrices: \((a - a, b - c, c - b, d - d)\). The resultant matrix \(\begin{pmatrix} 0 & b - c \ c - b & 0 \end{pmatrix}\) must equal \(\begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}\). Thus, we get the conditions: \(b - c = 0\) and \(c - b = 0\), which both simplify to \(b = c\).

Step 5: Final Condition

From the above equations, the condition that must be satisfied for the original equation \(A - A^{\prime} = \mathbf{0}\) to hold is \(b = c\), meaning that the matrix \(A\) must be symmetric.

Key Concepts

Matrix Transposition

Matrix transposition is a fundamental concept in matrix algebra. When we transpose a matrix, we simply swap its rows and columns. This means that the element located at the

• first row (i) and second column (j) in the original matrix will be moved to the second row (j) and first column (i) in the transposed matrix.
• This action is denoted by either a prime symbol (\(^\prime\)) or a capital T, so the transpose of a matrix \(A\) is commonly written as \(A^{\prime}\) or \(A^T\).

For example, consider a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\). The transpose of \(A\), written as \(A^{\prime}\), would be \(\begin{pmatrix} a & c \ b & d \end{pmatrix}\). Notice how the positions of \(b\) and \(c\) swap in the transposition.
Transposition is a simple but powerful tool because:

• it helps in solving equations,
• finding symmetries, and
• transforming problems into a more workable form.

Symmetric Matrix

A symmetric matrix is a square matrix that remains unchanged when transposed. This means that for a matrix \(A\) to be symmetric, it must equal its transpose (\(A = A^T\)). A key feature of symmetric matrices is that their elements satisfy certain conditions:

• The element at the i-th row and j-th column must equal the element at the j-th row and i-th column.

For a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \ b & d \end{pmatrix}\), it is symmetric because \(A^T = \begin{pmatrix} a & b \ b & d \end{pmatrix}\), which is identical to \(A\).
Symmetric matrices are significant in many areas of mathematics and statistics. They:

• often represent systems with inherent balance or equilibrium, and
• are easier to compute eigenvalues and eigenvectors for, simplifying many mathematical models.

In the original exercise, for \(A - A^\prime = \mathbf{0}\) to hold, \(A\) must be symmetric, implying that \(b = c\).

Zero Matrix

The zero matrix is a matrix where all its elements are zero. It is often represented as \(\mathbf{0}\) and serves a vital role as the additive identity in matrix algebra.
For instance, a \(2 \times 2\) zero matrix looks like \(\begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}\). In the context of solving matrix equations, like \(A - A^\prime = \mathbf{0}\), the zero matrix represents the result we obtain when the matrix \(A\) is subtracted from its transpose and yields no residual - an indication of symmetry in the matrix.

• This property is used to find when two matrices are identical and to ensure matrix operations conform to specific identities.
• The zero matrix is pivotal in illustrating concepts like null spaces or kernel of a matrix, where the linear transformation yields a result of all zeros.

Overall, the zero matrix simplifies the resolution of matrix equations, proving especially useful when verifying if one matrix is the inverse or transpose of another.