MHB Which of the Following is Incorrect Regarding Matrices and Vectors?

[Yankel]

One last question on these topics, I need to choose the WRONG statement, and they all seem correct to me...

a) If A is a squared matrix for which
\[A^{2}-A=0\]

then A=0 or A=i

b) If A and B are diagonal matrices, then Ab=BA

c) A 4X4 matrix with eigenvalues 1,0,-1,2 is "diagonlizable"

d) The dimension of the polynomials space of order 3 (ax^3+bx^2+...) is 4

e) If two vectors are linearly dependent, then one is necessarily a multiplication of the other

'a' is correct
'b', not sure, I tried one example, it worked
'c' Each eigenvalue appears once, so it's not possible to have an eigenvalue which appears twice with corresponding 1 eigenvector (for example)
'd' Isn't it 4 ?
'e' I think so...

will appreciate your help

 

[Ackbach]

Concerning answer a): consider
$$A=\begin{bmatrix} 1 &1 \\ 0 &0 \end{bmatrix}.$$

 

[Deveno]

To underscore a common mistake:

Just because:

$A^2 - A = A(A - I) = 0$

There is NO REASON to believe $A = 0$ or $A - I = 0$.

It is VERY POSSIBLE to have matrices $A,B$ with $AB = 0$ but $A,B \neq 0$.

Any such matrix, of course, is singular.