If `I=[[1,0],[0,1]]` and `E=[[0,1],[0,0]]` then show that `(al+bE)^3=a^(3)l+3a^(2)bE`

Show that [[1,1],[0,1]]^2 = [[1,3],[0,1]]

If A=[[3,1],[-1,2]], I=[[1,0],[0,1]] and O=[[0,0],[0,0]] , show that A^2-5A+7I=0 . Hence find A^(-1) .

If A=[[1 ,0 ,1],[0 ,1, 2],[ 0, 0, 4]] , then show that |3A| = 27|A|

If A=[[1 ,0 ,1],[0 ,1, 2],[ 0, 0, 4]] , then show that |3A|" "=" "27|A|

If A=[[3, 1],[-1, 2]] , show that A^2-5A+7I=0

If A=[[0,1],[1,0]],B=[[0,-i],[i,0]] and C=[[i,0],[0,-i]] , show that A^2=B^2=-C^2=I_2 and AB=-BA,AC=-CA and BC=-CB .

If A=[[1,2,3],[0,1,0],[1,1,0]] and B=[[-1,1,0],[0,-1,1],[2,3,4]] show that AB!=BA

(i) if A=[{:(1,0),(0,1):}],B=[{:(0,1),(1,0):}]and C=[{:(1,0),(0,1):}], then show that A^(2)=B^(2)=C^(2)=I_(2). (ii) if A=[{:(1,0),(1,1):}],B=[{:(2,0),(1,1):}]and C=[{:(-1,2),(3,1):}], then show that A(B+C)=AB+AC. (iii) if A=[{:(1,-1),(-1,1):}]and B=[{:(1,1),(1,1):}], then show that AB is a zero matrix.

If A=[[4,-1,-4],[ 3, 0,-4],[ 3,-1,-3]] , show that A^2=I_3dot

If (a+1/a)^2=3 and ane0 then show that a^3+1/a^3=0