Let A and B be n times n matrices. Recall that the trace of A , written tr(A),equal sum_{i=1}^nA_{ii} Prove that tr(AB)=tr(BA) and tr(A)=tr(A^t)

Question

Let A and B be

n \times n

matrices. Recall that the trace of A , written tr(A),equal

\sum_{i = 1}^{n} A_{i i}

Prove that tr(AB)=tr(BA) and

t r (A) = t r (A^{t})

Answer 1

Step 1
Consider A and B be

n \times n

matrices and trace of matrix is defined as,

t r (A) = \sum_{i = 1}^{n} A_{i i}

step 2
To prove tr(AB)=tr(BA).

t r (A B) = \sum_{i = 1}^{n} (A B)_{i i}

= \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i i} b_{j j}

= \sum_{j = 1}^{n} \sum_{i = 1}^{n} b_{j j} a_{i i}

= \sum_{i = 1}^{n} (B A)_{i i}

= t r (B A)

Hence, it is proved
Step 3
To prove

t r (A) = t r (A^{t})

Let

A = (a_{i i}), 1 \leq i \leq n and A^{T} = (b_{i i})

Then,

b_{i i} = a_{i i}

t r (A) = \sum_{i = 1}^{n} (A)_{i i}

= \sum_{i = 1}^{n} a_{i i}

= \sum_{i = 1}^{n} b_{i i}

= \sum_{i = 1}^{n} (A^{T})_{i i}

= t r (A^{T})

Hence, it is proved

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