Let N = [0 1 2; 0 0 3; 0 0 0] and let I be the 3 × 3 identity. What is the top right entry of (I + N)⁵, that is, the entry in row 1, column 3?

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A power that stops growing

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Let $N = 000100230$ and let $I$ be the $3 \times 3$ identity. What is the top right entry of $(I + N)^{5}$ , that is, the entry in row $1$ , column $3$ ?

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Hint

Compute a few powers of $N$ first and see how soon they vanish.

Solution

Since $N$ is strictly upper triangular, $N^{2} = 000000300$ and $N^{3} = 0$ .
The matrices $I$ and $N$ commute, so the binomial theorem applies and stops early: $(I + N)^{5} = I + 5 N + 10 N^{2}$ .
The entry in row $1$ , column $3$ is $0 + 5 \times 2 + 10 \times 3 = 40$ .

Answer: $40$

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