Question

If A is an invertible matrix of order 3, then which of the following is not true ?

पर्याय

• \[\left| adj A \right| = \left| A \right|^2\]

• \[\left( A^{- 1} \right)^{- 1} = A\]

• If \[BA = CA,\text{ than }B \neq C\] , where B and C are square matrices of order 3

• \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} , where B \neq \left[ b_{ij} \right]_{3 \times 3} and \left| B \right| \neq 0\]

Answer 1

If \[BA = CA\] , then \[B \neq C\]  where B and C are square matrices of order 3.

If A is an invertible matrix, then \[A^{- 1}\] exists. 
Now,

\[BA = CA\]

On multiplying both sides by \[A^{- 1}\]

\[A^{- 1}\] \[BA A^{- 1} = CA A^{- 1}\]

\[\Rightarrow BI = CI ..............\left[ \because A A^{- 1} =\text{ I where I is the identity matrix }\right]\]

\[ \Rightarrow B = C\]

Therefore, If \[BA = CA\] , then \[B \neq C\]  where B and C are square matrices of order 3 is not true.