The trace of an n ×n matrix A ∈ℳn ×n is defined to be the sum of its diagonal entries: tr A=a11+

step 1 So in the given question we are told to find the trace of the matrix that has been given to us.

The trace of an n ×n matrix A ∈ℳn ×n is defined to be the...

The trace of an $n \times n$ matrix $A \in \mathcal{M}_{n \times n}$ is defined to be the sum of its diagonal entries: $\operatorname{tr} A=a_{11}+a_{22}+\cdots+a_{n n}$. Prove that the set of trace zero matrices, $\operatorname{tr} A=0$, is a subspace of $\mathcal{M}_{n \times n}$.

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Key Concepts

Trace Operator

The trace of a matrix is the sum of its diagonal entries. This operator is a linear function from the space of n x n matrices to the underlying field, meaning that it satisfies the properties of additivity and homogeneity. It is fundamental in various areas of linear algebra, including the study of matrix invariants and spectral theory.

Vector Subspace Criteria

A subspace is a subset of a vector space that itself is a vector space under the same operations. To verify that a set is a subspace, one must show that it contains the zero vector, and that it is closed under both vector addition and scalar multiplication. This concept is critical for understanding the structure of vector spaces and the behavior of linear transformations.

Matrix Operations and Closure

Matrix addition and scalar multiplication are defined entrywise, and they satisfy the usual axioms of a vector space. In the context of trace zero matrices, these operations interact with the trace operator in a way that preserves the property of having a zero trace. Specifically, the additivity and homogeneity of the trace ensure that the sum or scalar multiple of trace zero matrices remains trace zero, confirming the closure properties required for a subspace.

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