If A= [ array ccc 1 & 1 & 3 5 & 2 & 6 -2 & -1 & -3 array ] , then A+A^3+A^4+A^5+3 I=

Step-by-step solution

A= [ array ccc 1 & 1 & 3 5 & 2 & 6 -2 & -1 & -3 array ] Its characteristics equation is: |A- I|=0 aligned & | array ccc 1- & 1 & 3 5 & 2- & 6 -2 & -1 & -3- array |=0 & (1- ) [ ^2+ -6+6 ]-1[15-5 +12] & +3[-5+4-2 ]=0 & .- ^3=0 ^2 )+3+5 -3-6 =0 aligned Every matrix satisfies its characteristics equation. A ^3=0 Now, A+A^3+A^4+A^5+3 I aligned & =A+0+A^3 A+A^3 A^2+3 I & =A+0+0+0+3 I & = [ array ccc 1 & 1 & 3 5 & 2 & 6 -2 & -1 & -3 array ]+3 [ array lll 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 array ]= [ array ccc 4 & 1 & 3 5 & 5 &