Answer to Question #219072 in Linear Algebra for Unknown346307

Question

Let A and B be n × n matrices. Prove that trAB = trBA and trA = trA^t

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Answer 1

A and B are matrices of order $n$ .

We know trace is sum of diagonal elements of the matrix.

We need to prove $tr$ ($AB$ ) = $tr$ ($BA$ )

When we multiply matrix A with B i.e AB or when we multiply matrix B with A i.e BA in both cases elements in the diagonal will remain the same. So sum of diagonal elements in AB =sum of diagonal elements in BA .

So $tr$ ($AB$ ) = $tr$ ($BA$ ) (proved)

Now we need to prove $tr$ ($A^t)=tr(A)$

So in $A^t$ diagonal elements will remain same as that of A .So sum of diagonal elements in A=sum of diagonal elements in $A^t$

$\implies tr(A^t)=tr(A)$ (proved)