JEE Main 2021 · 25 July · evening · Single correct · +4 / −1
If P = [ 1, 0; 1/2, 1 ], then P⁵⁰ is :
- (a) [ 1, 0; 25, 1 ]
- (b) [ 1, 50; 0, 1 ]
- (c) [ 1, 25; 0, 1 ]
- (d) [ 1, 0; 50, 1 ]
Answer and solution
Answer (a) [ 1, 0; 25, 1 ]
Solution
Maths sir · 4 steps
Split P into I plus a matrix whose square is zero, and the 50th power becomes easy.
- Split the matrix. Write P = I + N, where N = (0, 0; 1/2, 0). Squaring gives N² = (0, 0; 0, 0), so every higher power of N is also zero.
- Use the binomial expansion. I commutes with N, so the binomial theorem applies to (I + N)ⁿ. Every term with N² or a higher power of N vanishes, and only two terms survive.
Pⁿ = I + nN = (1, 0; n/2, 1)
- Check quickly. P² = (1, 0; 1, 1), and 2/2 = 1 matches the formula. Each multiplication by P adds 1/2 to the bottom-left entry.
- So. With n = 50, the bottom-left entry is 50/2 = 25, so P⁵⁰ = (1, 0; 25, 1). That is option (a).
The idea to keep. If a matrix is I + N with N² = O, then (I + N)ⁿ = I + nN. The off-diagonal entry grows linearly with n; it is never raised to a power.
Answer from the NTA final answer key. Solution by PrepWiser.