Question

If A and B are 2 matrices given by general elements $a_{ij}$ and $b_{ij}$ respectively. And $b_{ij}$ = Im($a_{ij}$). Then which the following are correct if A* + B = 0. ( Im(x) means Imaginary part of the number x)

• A. A is a symmetric matrix
• B. A is a non symmetric matrix
• C. Sum of real parts of elements of A is zero
• D. Sum of imaginary parts of elements of A is zero

Answer 1

Answer: (A), (C)

The correct options are
A

A is a symmetric matrix

C

Sum of real parts of elements of A is zero

Given matrices are A and B such that $b_{ij}=Im\left(a_{ij}\right)$. This simply means B’s elements are the imaginary part of elements of corresponding elements of A.

Its given that,

A* + B = 0.

At an elemental level this can be written as,

$\overline{aij}+bji=0$

i.e., Re$\left(a_{ij}\right)$ – Im$\left(a_{ij}\right)$ + Re $\left(b_{ji}\right)$ + Im$\left(b_{ji}\right)$=0 (1)

But its given that,

$b_{ij}$= Im$\left(a_{ij}\right)$ (2)

$\to Re\left(b_{ji}\right)=0$ (3)

Using (2) and (3) in (1)

Re$\left(a_{ij}\right)$ – Im$\left(a_{ij}\right)$ + 0 + Im $\left(a_{ji}\right)$=0.

Equating Real and imaginary parts separately.

Re$\left(a_{ij}\right)$=0

$\sum Re\left(a_{ij}\right)=0$

This shows elements in A are purely imaginary.

Also, Im$\left(a_{ij}\right)$ = Im $\left(a_{ji}\right)$.

i.e., $a_{ij}$= $a_{ji}$

which shows A is symmetric.

Hence both options (a) & (c) are correct