91Ó°ÊÓ

Use partitioning to compute the inverses of the following matrices: (a) \(\left(\begin{array}{lllll}1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1\end{array}\right)\) (b) \(\left(\begin{array}{llll}2 & 3 & 0 & 0 \\ 5 & 2 & 0 & 0 \\ 0 & 0 & 4 & 3 \\ 0 & 0 & 3 & 2\end{array}\right)\) (c) \(\left(\begin{array}{llll}2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\end{array}\right)\)

Suppose \(\mathbf{A}\) is a square matrix and let \(\lambda\) be an eigenvalue of \(\mathbf{A}\). Prove that if \(|\mathbf{A}| \neq 0\), then \(\lambda \neq 0\). In this case show that \(1 / \lambda\) is an eigenvalue of the inverse \(\mathbf{A}^{-1}\).

Let \(\mathbf{A x}=\mathbf{b}\) be a linear system of equations in matrix form. Prove that if \(\mathbf{x}_{1}\) and \(\mathrm{x}_{2}\) are both solutions of the system, then so is \(\lambda \mathbf{x}_{1}+(1-\lambda) \mathbf{x}_{2}\) for every real number \(\lambda\). Use this fact to prove that a linear system of equations that is consistent has either one solution or infinitely many solutions. (For instance, it cannot have exactly 3 solutions.)

Cayley-Hamilton's theorem can be used to compute powers of matrices. In particular, if \(\mathbf{A}=\left(a_{i j}\right)\) is \(2 \times 2\), then $$ \mathbf{A}^{2}=\operatorname{tr}(\mathbf{A}) \mathbf{A}-|\mathbf{A}| \mathbf{I} $$ Multiplying this equation by \(\mathbf{A}\) and using \((*)\) again yields \(\mathbf{A}^{3}\) expressed in terms of \(\mathbf{A}\) and \(\mathbf{I}\), etc. Use this method to find \(\mathbf{A}^{4}\) when \(\mathbf{A}=\left(\begin{array}{ll}2 & 1 \\ 1 & 3\end{array}\right)\). (See e.g. Goldberg (1958), Section 4.5.)

Prove that \((1,1,1),(2,1,0),(3,1,4)\), and \((1,2,-2)\) are linearly dependent.