Question

Prove that every diagonal matrix is a symmetric matrix. Is the converse true ?
Justify.​

Answer 1

$\huge\mathbb\purple{BONJOUR}$

$\huge\mathbb\pink{ANSWER:}$

You can't use the thing you want to prove in the proof itself, so the above answers are missing some steps. Here is a more complete proof. Given A is nonsingular and symmetric, show that A−1=(A−1)T:

I=IT

since AA−1=I,

AA−1=(AA−1)T

since (AB)T=BTAT,

AA−1=(A−1)TAT

since AA−1=A−1A=I, we rearrange the left side

A−1A=(A−1)TAT

since A=AT, we substitute the right side

A−1A=(A−1)TA

A−1A(A−1)=(A−1)TA(A−1)

A−1I=(A−1)TI

A−1=(A−1)T

and we are done.

$\huge\mathbb\green{@Naira}$