Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix?

Problems and solutions in Linear Algebra

Problem 633

Let $A$ be an $n \times n$ matrix.

Is it true that $\tr ( A^\trans ) = \tr(A)$? If it is true, prove it. If not, give a counterexample.

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The answer is true. Recall that the transpose of a matrix is the sum of its diagonal entries. Also, note that the diagonal entries of the transposed matrix are the same as the original matrix.

Putting together these observations yields the equality $\tr ( A^\trans ) = \tr(A)$.

Here is the more formal proof.

For $A = (a_{i j})_{1 \leq i, j \leq n}$, the transpose $A^{\trans}= (b_{i j})_{1 \leq i, j \leq n}$ is defined by $b_{i j} = a_{j i}$.

In particular, notice that $b_{i i} = a_{i i}$ for $1 \leq i \leq n$. And so,
\[ \tr(A^{\trans}) = \sum_{i=1}^n b_{i i} = \sum_{i=1}^n a_{i i} = \tr(A) . \]

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