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Verify Cayley-Hamilton theorem for following matrix and also find the following

A1andA4

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Characteristic equation of A is AλI∣=0

λ36λ2+[(4+4+4)(1+1+1)]λA∣=0

λ36λ2+(123)λ4=0

λ36λ2+9λ4=0

To verify Cayley-Hamilton Theorem, we need to prove that A36A24I=0

LHS=A36A2+9A4I

= [222121 212221 212122 ]-6[655565556]+9[211121112] - [400040004]

= [222121 212221 212122 ]-6[363030303630303036]+9[189991899918]

= [222121 212221 212122 ]-6[655565556]+9[211121112] - [400040004]

= [000 000 000 ]= RHS A36A2+9A4I=0 (1) Cayley-Hamilton Theorem verified. Now to find A1 and A4. multiply (1) by A1 we get, A1(A36A2+9A4I)=A10 A26A+9I4A1=0 4A1=A26A+9I A1=14(A26A+9I) A1=[655565556] 6[211121112] + [900090009]]

[655565556] -[126661266612] + [900090009]]

=[311131113]

To find A4 multiply (1) by A

A(A36A2+9A4I)=A(0)

A46A3+9A24A=0

A4=6A39A2+4A

=6[222121212221212122] 9[655565556] +4 [211121112]

=[132126126126132126126126132] [544545455445454554] + [844484448]

=[868585858685858586]

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