Question 16.6.4: Let B be a n × n matrix such that B² = 3B. Prove that there ......
Let B be a n × n matrix such that B² = 3B. Prove that there exists a number s such that I + sB is the inverse of I + B.
Step-by-Step
Because of (16.6.5), it suffices to find a number s such that (I + sB)(I + B) = I. Now,
YA = I ⇒ Y = A^{−1} (16.6.5)
(I + sB)(I + B) = II + IB + sBI + sB² = I + B + sB + 3sB = I + (1 + 4s)B
which is equal to I provided 1 + 4s = 0. The right choice of s is, thus, s = −1/4.
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