Question

If A is square matrix, then which of the following is not true ? OPTIONS --- 1) AA' is symmetric 2) A-A' is skew symmetric. 3) A square is symmetric 4) A+A' is symmetric​

Answer 1

Answer:

option (3) is not true

Step-by-step explanation:

Concept used:

A square matrix A is said to be symmetric if A=A'

A square matrix A is said to be skew symmetric if A= - A'

Properties of transpose:

$\boxed{\begin{minipage}{4cm} 1.\:(AB)'=B'A'\\ \\2.(A')'=A\\ \\3.(A+B)'=A'+B'\\ \\4.(A-B)'=A'-B' \end{minipage}}$

1.

$(AA')'$

$=(A')'A'$

$=AA'$

$\implies\,(AA')'=AA'$

$\therefore\,\text{AA' is symmetric matrix}$

2.

$(A-A')'$

$=A'-(A')'$

$=A'-A$

$=-(A-A')$

$\implies\,(A-A')'=-(A-A')$

$\therefore\,\text{A-A' is skew symmetric matrix}$

4

$(A+A')'$

$=A'+(A')'$

$=A'+A$

$=A+A'$

$\implies\,(A+A')'=A+A'$

$\therefore\,\text{A+A' is symmetric matrix}$