question_answer

If and if A is invertible, then which of the following is not true?

A)  \[\left| A \right|=\left| B \right|\]

B)  \[\left| A \right|=-\left| B \right|\]

C)  \[\left| adj\,A \right|=\left| adj\,B \right|\]

D)  A is invertible if and only if B is invertible

Correct Answer: A

Solution :

[a] (Multiplying \[{{R}_{2}}\]by -1)  (Multiplying \[{{C}_{2}}\]by -1)  (Changing \[{{R}_{1}}\]with\[{{R}_{2}}\]) Hence, \[\left| A \right|=-\left| B \right|,\] obviously when \[\left| A \right|\ne 0,\left| B \right|\ne 0.\] Also, \[\left| adjB \right|={{\left| B \right|}^{2}}={{(-\left| A \right|)}^{2}}={{\left| A \right|}^{2}}.\]