Let I be an identity matrix of order 2 × 2 and P = [ 2, - 1; 5, - 3 ]. Then the value of n∈N for which Pⁿ = 5I - 8P is equal to ____________.

Solution

Maths sir · 3 steps

Do not multiply matrices six times: get P² in terms of P and I, then keep reducing.

1. Find P squared. For P, the trace is 2 + (-3) = -1 and |P| = -6 + 5 = -1. By the Cayley–Hamilton theorem, P² - (trace)P + |P| I = O.

   P² + P - I = O ⇒ P² = I - P

2. Climb the powers. Multiply by P each time and replace every P² with I - P. Each power then stays of the form aI + bP.

   Power	Working	Result
   P³	P - P² = P - (I - P)	2P - I
   P⁴	2P² - P = 2I - 2P - P	2I - 3P
   P⁵	2P - 3P² = 2P - 3I + 3P	5P - 3I
   P⁶	5P² - 3P = 5I - 5P - 3P	5I - 8P

3. So. P⁶ = 5I - 8P, so n = 6.

The idea to keep. For any 2 × 2 matrix, P² = (trace)P - |P| I. Use it to write every higher power as aI + bP, and then match coefficients.

Answer from the NTA final answer key. Solution by PrepWiser.