Chapter 2.1, Problem 7P

Explanation of Solution

a.

Trace of the given matrix:

The trace of a matrix is the sum of its diagonal elements.

Now, consider a matrix $A$ of order $n\times n$ having elements $a_{ij}$ and another matrix $B$ of same order having elements $b_{ij}$.

Now, the addition of these matrices is,

$A+B=\left[a_{ij}+b_{ij}\right]$

The trace of this matrix is given below:

$trace\left(A+B\right)={\displaystyle \sum _{i=1}^n\left[a_{ii}+b_{ii}\right]}$

We can write this as follows:

  $$

Explanation of Solution

b.

Proof:

Consider any $n\times n$ matrices $A$ and $B$.

Suppose $C=AB$.

Then the elements of this matrix are $c_{ij}$.

Then we can write,

$c_{ij}={\displaystyle \sum _{k=1}^n\left[a_{ik}{b}_{kj}\right]}$

Similarly, let $D=BA$.

Then the elements of this matrix are $d_{ij}$.

Then we can write,

$d_{ij}={\displaystyle \sum _{k=1}^n\left[b_{ik}{a}_{kj}\right]}$

Now, the trace of the matrix $AB$ is given below:

$trace\left(AB\right)={\displaystyle \sum _{i=1}^n{c}_{ii}}$