- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.16 Exercises
8. $\quad$ Let $A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}.$ Verify that $A^2 = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}.$ Compute $A^3$ and $A^4.$ Guess a general formula for $A^n$ and prove it by induction.
Solution. $\quad$ By integral powers of square matrices, we have \begin{align*} A^2 &= \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix} \\ \\ A^3 &= \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} 1 & 3 & 6 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{bmatrix} \\ \\ A^4 &= \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 6 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} 1 & 4 & 10 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix} \end{align*} This sequence of matrices suggests the following general formula for $n \geq 1$ \begin{align*} A^n &= \begin{bmatrix} 1 & n & \sum_{k=1}^n k \\ 0 & 1 & n \\ 0 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} 1 & n & \frac{n(n+1)}{2} \\ 0 & 1 & n \\ 0 & 0 & 1 \end{bmatrix} \end{align*} Assume the above equation holds for some integer $n \geq 1,$ we will now show that it also holds for $n + 1:$ \begin{align*} A^{n+1} &= \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & n & \sum_{k=1}^n k \\ 0 & 1 & n \\ 0 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} (1 + 0 + 0) & (n + 1 + 0) & \left[\left(\sum_{k=1}^n k\right) + n + 1\right] \\ (0 + 0 + 0) & (0 + 1 + 0) & (0 + n + 1) \\ (0 + 0 + 0) & (0 + 0 + 0) & (0 + 0 + 1) \end{bmatrix} \\ &= \begin{bmatrix} 1 & (n + 1) & \sum_{k=1}^{n+1} k \\ 0 & 1 & (n + 1) \\ 0 & 0 & 1 \end{bmatrix} \end{align*} But since this equation holds for $n + 1,$ by induction, it holds for all integers $\geq 1. \quad \blacksquare$