A is a 3x3 matrix, and 0 ≠ a ^ 3 = a ^ 2 ≠ a, 1). It is proved that a is not diagonalizable. 2.) 0 is the eigenvalue of A. 3). 1 is the eigenvalue of A A is a 3x3 matrix, and 0 ≠ a ^ 3 = a ^ 2 ≠ a, 1) 2.) 0 is the eigenvalue of A 3) . 1 is the eigenvalue of A 4) Give an example of a, which needs to satisfy the condition a ^ 3 = a ^ 2 ≠ a 5) Prove that any 2x2 matrix does not satisfy the condition a ^ 3 = a ^ 2 ≠ a

A is a 3x3 matrix, and 0 ≠ a ^ 3 = a ^ 2 ≠ a, 1). It is proved that a is not diagonalizable. 2.) 0 is the eigenvalue of A. 3). 1 is the eigenvalue of A A is a 3x3 matrix, and 0 ≠ a ^ 3 = a ^ 2 ≠ a, 1) 2.) 0 is the eigenvalue of A 3) . 1 is the eigenvalue of A 4) Give an example of a, which needs to satisfy the condition a ^ 3 = a ^ 2 ≠ a 5) Prove that any 2x2 matrix does not satisfy the condition a ^ 3 = a ^ 2 ≠ a

Suppose a is diagonalizable, let p-1ap = diag (a, B, c). (1) (diagonal matrix of a, B, C on diagonal) p be invertible
(1) The square of two sides: p-1a ^ 2p = diag (a ^ 2, B ^ 2, C ^ 2). (2)
(1) P-1a ^ 3P = diag (a ^ 3, B ^ 3, C ^ 3). (3)
From (2) (3) and a ^ 2 = a ^ 3, so a ^ 2 = a ^ 3, B ^ 2 = B ^ 3, C ^ 2 = C ^ 3
So a, B, C belong to the set {0,1}, so a = a ^ 2 = a ^ 3
In this case, diag (a, B, c) = diag (a ^ 2, B ^ 2, C ^ 2) = diag (a ^ 3, B ^ 3, C ^ 3), and a = P diag (a, B, c) P-1
A ^ 2 = P diag (a ^ 2, B ^ 2, C ^ 2) P-1, then a = a ^ 2 contradiction!
So a is not diagonalizable
That is, there is a vector α such that a α = 0 & nbsp; & nbsp; that is, a is irreversible
Proof to the contrary: suppose that a is reversible and there is an inverse a of a & nbsp; - 1, then in the formula A ^ 3 = a ^ 2, multiply A-1 left to get: A ^ 2 = a, contradiction
Prove that A-I is irreversible (I is the identity matrix)
The same counter argument: suppose that A-I is invertible, let its inverse matrix be (A-I) - 1
Then a ^ 3-A ^ 2 = 0, so a ^ 2 (A-I) = 0
Right multiply (A-I) - 1: A ^ 2 = 0
It is contradictory to the problem condition that 0 is not equal to a ^ 3 = a ^ 2
example:
010
000
001
The topic should be to prove that any 2x2 matrix does not satisfy the condition 0 ≠ a ^ 3 = a ^ 2 ≠ a, otherwise
01
00 is a counterexample
It is proved that any 2x2 matrix does not satisfy the condition 0 ≠ a ^ 3 = a ^ 2 ≠ a
Counter proof: suppose that all 2x2 matrices satisfy the condition 0 ≠ a ^ 3 = a ^ 2 ≠ a, similarly: the matrix cannot be diagonalized, and 0 and 1 are eigenvalues
Because 0 and 1 are two different eigenvalues, the 2x2 matrix can be diagonalized
So we have to