Problem 33 - Differential Equations and Linear Algebra

Question

Show that if \(A\) is an \(n \times n\) matrix that is both symmetric and skew- symmetric, then every element of \(A\) is zero. (Such a matrix is called a zero matrix.)

Step-by-Step Solution

Answer

Given an \(n\times n\) matrix \(A\), if \(A\) is both symmetric (\(A^T=A\)) and skew-symmetric (\(A^T=-A\)), we have \(a_{ij}=a_{ji}\) and \(a_{ij}=-a_{ji}\) for all elements in the matrix. Combining these conditions, we get \(a_{ij}=-a_{ij}\), resulting in \(2a_{ij}=0\) for all elements. Consequently, every element of \(A\) is zero, making \(A\) a zero matrix.

Step 1: Definition of a Symmetric Matrix

A symmetric matrix is a square matrix that satisfies the condition \(A^{T} = A\), where \(A^{T}\) denotes the transpose of matrix \(A\). In other words, the element in the ith row and jth column of a symmetric matrix must be equal to the element in the jth row and ith column: \(a_{ij} = a_{ji}.\)

Step 2: Definition of Skew-symmetric Matrix

A skew-symmetric matrix is a square matrix that satisfies the condition \(A^{T} = -A\), where \(A^{T}\) is the transpose of matrix \(A\). This means that the element in the ith row and jth column of a skew-symmetric matrix must be equal to the negative of the element in the jth row and ith column: \(a_{ij} = -a_{ji}.\)

Step 3: Comparing Symmetric and Skew-symmetric Conditions

Now, let's assume we have a matrix A which is both symmetric and skew-symmetric. By the definitions given in Steps 1 and 2, this means that both \(a_{ij} = a_{ji}\) and \(a_{ij} = -a_{ji}\) must be satisfied for all elements in the matrix.

Step 4: Find Each Element of A to be Zero

Since both \(a_{ij} = a_{ji}\) and \(a_{ij} = -a_{ji}\) must hold true, we can write the equation as \(a_{ij} = -a_{ij}\) for all elements in matrix A. By adding \(a_{ij}\) to both sides of the equation, we find that \(2a_{ij} = 0\). Thus, every element of A must be zero: \(a_{ij} = 0\).

Step 5: Conclusion

Since all elements in matrix A are zero, the matrix A must be a zero matrix. Therefore, we have proved that if A is an n x n matrix that is both symmetric and skew-symmetric, then every element of A is zero, and A is a zero matrix.

Key Concepts

Symmetric Matrix

A symmetric matrix is a specific type of square matrix where each element mirrors itself across the main diagonal. To understand it better, imagine the main diagonal running from the top left corner to the bottom right corner. In a symmetric matrix, the element located at position (i, j) is identical to the element at position (j, i). This property can be mathematically expressed as:

• \(A^T = A\) which means the matrix is unchanged when transposed.
• This implies that \(a_{ij} = a_{ji}\) for all i and j.

The condition of symmetry is very useful in many mathematical applications because it often simplifies computations and analysis. Additionally, symmetric matrices appear frequently in various fields such as physics, statistics, and computer science. They help represent symmetric relationships, like distances or correlations.

Skew-Symmetric Matrix

A skew-symmetric matrix has a unique property that differentiates it from a symmetric matrix. It is also a square matrix, but here the transpose of the matrix is equal to the negative of the original matrix:

• \(A^T = -A\), which means swapping rows and columns gives us negative values of the original elements.
• This leads to the equation: \(a_{ij} = -a_{ji}\) for all i and j.

An interesting aspect of skew-symmetric matrices is that the elements along the main diagonal must be zero. This is because the condition \(a_{ii} = -a_{ii}\) can only be satisfied if \(a_{ii} = 0\). Skew-symmetric matrices are commonly found in areas dealing with alternating relationships or vectors, such as in electromagnetic fields or rotational dynamics. They are particularly important in operations involving determinants and eigenvalues.

Zero Matrix

Both symmetric and skew-symmetric conditions seem conflicting at first, but they lead to important conclusions. When combined, these conditions dictate that a matrix must have all zero elements. To see why, consider the conditions for symmetry and skew-symmetry:

• Symmetric: \(a_{ij} = a_{ji}\)
• Skew-symmetric: \(a_{ij} = -a_{ji}\)

These combined imply that \(a_{ij} = -a_{ij}\), meaning \(2a_{ij} = 0\), thus \(a_{ij} = 0\) for all elements. Therefore, when a matrix is both symmetric and skew-symmetric, every element equals zero, describing a zero matrix. A zero matrix is an important type of matrix where every element is zero. It acts as a neutral element in matrix addition, meaning adding any matrix to a zero matrix results in the original matrix remaining unchanged. Zero matrices are vital in linear algebra for representing zero transformations or as additive identities.