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

A is a symmetric matrix

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A is a non symmetric matrix

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Sum of real parts of elements of A is zero

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Sum of imaginary parts of elements of A is zero

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Solution

The correct option is C
Sum of real parts of elements of A is zero

Given matrices are A and B such that $b_{i j} = I m (a_{i j})$ . 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,

$¯ ¯¯¯¯¯ ¯ a i j + b j i = 0$

i.e., Re $(a_{i j})$ – Im $(a_{i j})$ + Re $(b_{j i})$ + Im $(b_{j i})$ =0 (1)

But its given that,

$b_{i j}$ = Im $(a_{i j})$ (2)

$\to R e (b_{j i}) = 0$ (3)

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

Re $(a_{i j})$ – Im $(a_{i j})$ + 0 + Im $(a_{j i})$ =0.

Equating Real and imaginary parts separately.

Re $(a_{i j})$ =0

$\sum R e (a_{i j}) = 0$

This shows elements in A are purely imaginary.

Also, Im $(a_{i j})$ = Im $(a_{j i})$ .

i.e., $a_{i j}$ = $a_{j i}$

which shows A is symmetric.

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

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