Suppose a matrix A satisfies A2 -5A +7I =0 If A5 = aA +bI then find the - Maths - Matrices

Question

Suppose a matrix A satisfies A2 -5A +7I =0 If
A5 = aA +bI then find the value of 2a -3b

Akhil Goyal · Accepted Answer

Given:A2-5A+7=0   -----(1)⇒A2=5A-7    -----(2)Multiplying (1) by A on both sides:A3-5A2+7A=0Substituting (2):A3-5(5A-7)+7A=0⇒A3-18A+35=0Again multiplying by A on both sides:A4-18A2+35A=0again, substituting (2):A4-18(5A-7)+35A=0⇒A4-55A+126=0Again multiplying by A on both sides and substuting (2):A5-55(5A-7)+126A=0A5-149A+385=0⇒A5=149A-385a=149, b=(-385)2a-3b=298+1155=1453

Suppose a matrix A satisfies A2 -5A +7I =0 .If A5 = aA +bI . then find the value of 2a -3b

Suppose a matrix A satisfies A² -5A +7I =0 .If A⁵ = aA +bI . then find the value of 2a -3b