Let \(A = \begin{pmatrix}1 & -1 & 0\\0 & 1 & -1\\0 & 0 & 1\end{pmatrix}\) and \(B = 7A^{20} - 20A^{7} + 2I\), where I is an identity matrix of order \(3 \times 3\). If \(B = [b_{ij}]\), then \(b_{13}\) is equal to _____________.
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Correct Answer: 910
Solution and Explanation
Since I and N commute, \(A^n = I + nN + \frac{n(n-1)}{2}N^2\).
\(B = 7A^{20} - 20A^7 + 2I = (7 - 20 + 2)I + (7\cdot 20 - 20\cdot 7)N + \left(7\cdot\frac{20\cdot 19}{2} - 20\cdot\frac{7\cdot 6}{2}\right)N^2\)
\(= -11I + 0\cdot N + (1330 - 420)N^2 = -11I + 910N^2\).
The (1, 3) entry of I is 0 and of \(N^2\) is 1, so \(b_{13} = 910\).
Answer: 910
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