Engineering
Mathematics
Inverse of a Matrix
Question

Let B=1    31    5 and A be a 2 × 2 matrix such that AB–1 = A–1 . If BCB–1 = A and C4 + αC2 + βI = 0, then 2β – α is equal to 

8

16

2

10

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Solution

AB–1 = A–1 ⇒ A2B–1 = I ⇒ A2 = B

& BCB–1 = A ⇒ BC = AB ⇒ ABC = A2B

⇒ BC.C = B2 ⇒ BC2 = B2 ⇒ B–1BC2 = B–1B2

⇒ C2 = B

C4 + αC2 + βI = 0 ⇒ B2 + αβ + β.I = 0 ...(i)

Ch. Equation: |B – xI| = 0 ⇒ 1x315x=0 ⇒ x2 – 6x + 5 – 3 = 0

⇒ B2 – 6B + 2I = 0 ...(ii)

from (i) & (ii) 

α = –6, β = 2

so 2β – α = 4 + 6 = 10

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