Let $A$ be a $4 \times 4$ invertible real matrix then which of the following is not true
1) the rows of $a$ form a basis of $\Bbb R^4$.
2) null space of $A$ contains only the $0$ vector
3)$A$ has 4 distinct eigenvalues
4) image of the linear transformation $x$ to $\mathbb R^4$ is $\mathbb R^4$
Since $A$ is invertible option 2 is always true. i am not sure about another options please help me .
Best Answer
Hint:
Options (1), (2), (4) are indeed true.
With respect to option (3): is it necessarily the case that $A$ has 4 distinct eigenvalues? (Think of the identity matrix...)