a) $A^2 \neq I_2$, $A^4 = I_2$
b) $A^2 =A$ all entries of $A$ are nonzero.
By the way, a and b are different questions. They are different matrices that I need to find.
How can I find a $2 x 2$ matrix $A$ that satisfy the following properties?
0
$\begingroup$
linear-algebra
matrices
-
0Please use appropriate formatting. Otherwise it is not clear what you are asking. – 2017-02-11
-
0See http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – 2017-02-11
2 Answers
2
Think of what $A$ must represent in terms of linear transformations.
$(a)$ Can you think of a transformation that is not the identity if you apply it twice, but that leaves things unchanged if you apply it four times? Hint: try rotations!
$(b)$ Can you think of a kind of transformation that leaves things unchanged if it's already been applied? Hint: try projections!
0
Hint for 1: the eigenvalues of $A$ must be $i$ and $-i$
Hint for 2: if $v$ is a norm $1$ vector, then $vv^T$ is a projection matrix