Let P be a 2 x 2 matrix such that P102 = 0. Thena)P2 = 0b)(I — P)2 = 0c)(I + P)2 = 0d)P = 0Correct answer is option 'A'. Can you explain this answer?

Question

Let P be a 2 x 2 matrix such that P102 = 0. Thena)P2 = 0b)(I &mdash; P)2 = 0c)(I + P)2 = 0d)P = 0Correct answer is option 'A'. Can you explain this answer?

Answer

To understand what the equation P^102 = 0 means, we need to understand matrix exponentiation. The exponent of a matrix represents the number of times the matrix is multiplied by itself.

a) P^2 represents P multiplied by itself. So, if P^102 = 0, it means that P multiplied by itself 102 times equals the zero matrix.

b) (I - P^2) represents the identity matrix (I) minus P^2. Since P^2 = 0 (from part a), then (I - P^2) simplifies to (I - 0) = I.

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