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Question
If A is an invertible matrix of order 3, then which of the following is not true ?
Options
\[\left| adj A \right| = \left| A \right|^2\]
\[\left( A^{- 1} \right)^{- 1} = A\]
If \[BA = CA,\text{ than }B \neq C\] , where B and C are square matrices of order 3
\[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} , where B \neq \left[ b_{ij} \right]_{3 \times 3} and \left| B \right| \neq 0\]
Solution
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.
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