Prove or disprove the following statements. I and 0 denote respectively the identity
and zero matrix of the same size as A.

(a) A system with three equations and two unknowns must be inconsistent.
(b) A system of n equations with n unknowns has at most n solutions.
(c) AandBare2×2matricessuchthatAB=0thenwemusthaveA=0orB=0
(d) If the system AX = 0 has infinitely many solutions, so does the system AX = b
for any choice of b.
(e) If A is a square matrix such that A2 −3A+2I = 0 then A−cI is invertible
whenever c ̸= 1 and c ̸= 2.

Please help, I am very lost!

Question

Prove or disprove the following statements. I and 0 denote respectively the identity
and zero matrix of the same size as A.

(a) A system with three equations and two unknowns must be inconsistent.
(b) A system of n equations with n unknowns has at most n solutions.
(c) AandBare2×2matricessuchthatAB=0thenwemusthaveA=0orB=0
(d) If the system AX = 0 has infinitely many solutions, so does the system AX = b
for any choice of b.
(e) If A is a square matrix such that A2 −3A+2I = 0 then A−cI is invertible
whenever c ̸= 1 and c ̸= 2.

Please help, I am very lost!

loser66 · Answer

@e.mccormick  you should be here

e.mccormick · Answer

Well, one way to disprove things is to show an example that contradicts the statement.

loser66 · Answer

those information is basic, right? need to understand? make question and get explanation, please

e.mccormick · Answer

Is this inconsistent or consistent:

$2x +y =4$

$x +y =3$

$2x +2y =6$

anonymous · Answer

Thanks I just got it!

e.mccormick · Answer

A lot of those are just applying the rules.  For each rule there is a proof.  So you just need to find each proof.