Vectors and Matrices | Part IA, 2008

A real 3×33 \times 3 matrix AA with elements AijA_{i j} is said to be upper triangular if Aij=0A_{i j}=0 whenever i>ji>j. Prove that if AA and BB are upper triangular 3×33 \times 3 real matrices then so is the matrix product ABA B.

Consider the matrix

A=(120011001)A=\left(\begin{array}{rrr} 1 & 2 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & -1 \end{array}\right)

Show that A3+A2A=IA^{3}+A^{2}-A=I. Write A1A^{-1} as a linear combination of A2,AA^{2}, A and II and hence compute A1A^{-1} explicitly.

For all integers nn (including negative integers), prove that there exist coefficients αn,βn\alpha_{n}, \beta_{n} and γn\gamma_{n} such that

An=αnA2+βnA+γnIA^{n}=\alpha_{n} A^{2}+\beta_{n} A+\gamma_{n} I

For all integers nn (including negative integers), show that

(An)11=1,(An)22=(1)n, and (An)23=n(1)n1\left(A^{n}\right)_{11}=1, \quad\left(A^{n}\right)_{22}=(-1)^{n}, \quad \text { and } \quad\left(A^{n}\right)_{23}=n(-1)^{n-1}

Hence derive a set of 3 simultaneous equations for {αn,βn,γn}\left\{\alpha_{n}, \beta_{n}, \gamma_{n}\right\} and find their solution.

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