Which of the following are subspaces of the vector space of n ×n matrices ℳn ×n ? The set of all

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Which of the following are subspaces of the vector space of...

Which of the following are subspaces of the vector space of $n \times n$ matrices $\mathcal{M}_{n \times n}$ ? The set of all (a) regular matrices; (b) nonsingular matrices; (c) singular matrices; (d) lower triangular matrices; (e) lower unitriangular matrices; $(f)$ diagonal matrices; $(g)$ symmetric matrices; $(h)$ skew-symmetric matrices.

Key Concepts

Triangular and Diagonal Matrices

Subsets of matrices defined by specific structural forms, such as lower triangular or diagonal matrices, often form subspaces because the zero matrix retains the structure and addition and scalar multiplication do not disrupt the triangular or diagonal pattern. However, care must be taken if the structure imposes fixed nonzero entries (like unitriangular matrices), as this requirement may violate closure properties.

Singular Matrices

Singular matrices are those that are not invertible. Although this set contains the zero matrix, it fails the closure properties; for example, the sum of two singular matrices can be non?singular, which prevents this set from being a subspace.

Symmetric Matrices

Symmetric matrices are those that satisfy the condition A = A^T. This set forms a subspace because the zero matrix is symmetric, and both addition and scalar multiplication preserve symmetry, meaning any linear combination of symmetric matrices remains symmetric.

Closure Properties

The concepts of closure under addition and scalar multiplication are essential for subspaces. Any two elements in the subset, when added together or scaled by any scalar, must produce an element still within the subset. Failure of closure in even one direction disqualifies the subset from being a subspace.

Vector Subspace

A vector subspace is a subset of a vector space that is itself a vector space under the same operations. This means it must contain the zero vector and be closed under addition and scalar multiplication. These criteria are the foundation for determining whether any subset qualifies as a subspace.

Invertible (Regular) Matrices

Invertible matrices, often called regular or nonsingular matrices, do not form a subspace because the set does not include the zero matrix and is not closed under scalar multiplication. Multiplying an invertible matrix by zero or by other scalars may result in a singular matrix, breaking the necessary closure property.

Skew?Symmetric Matrices

Skew?symmetric matrices satisfy A^T = -A, and they also form a subspace. The zero matrix meets this condition, and the property is preserved under addition and scalar multiplication, ensuring that the set remains closed under the operations of the vector space.

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