Question

show that the sum of the elements of principal diagonal of (A+B)=sum of the elements of principal diagonal A+sum of the elements of principal diagonal B.....A and B are matrix thank you in advance

Answer 1

They're both square matrices, yes?

Answer 2

Write \( A = (a_{ij}) \) where A is a n x n matrix.

The sum of the principal diagonal is
\[ a_{11} + a_{22} + .... + a_{nn} \]
and the some of the principal diagonal for B is a similar expression.

Now, the sum of the two matrices \( A + B = (a_{ij} + b_{ij}) \)
So it's diagonal elements are just \( a_{11} + b_{11}, a_{22} + b{22}, ... \)

Answer 3

So it should now be obvious why this the relation asked is true.

Answer 4

If you're having a hard time with this, write out generic 3x3 matrices for A and B and see how it works.

Answer 5

yes they are both square matrices yes i know how it works but i don't know to express it in general temrs

Answer 6

So technically the name for the sum of the principal diagonal is the trace. We write the trace of a n x n matrix A is
\[ tr(A) = \sum_{i=1}^n a_{ii} \]

Answer 7

As the matrix \( A + B = (a_{ij} + b_{ij}) \), the trace
\[ tr(A+B) = \sum_{i=1}^n (a_{ii} + b_{ii}) = \sum_{i=1}^n a_{ii} + \sum_{i=1}^n b_{ii} = tr(A) + tr(B) \]

Answer 8

so its the same thing with these tr(kA)=ktrA tr(A^T(transposed))=trA and tr(A*B)=tr(B*A)

Answer 9

Yes. Just be careful with the last one.

Answer 10

prove that if A and B are antisymetric matrix than these are antisymetric too A^T ,A+B,A-B and k*A .please