91Ó°ÊÓ

Partition each of the following matrices so that it becomes a square block matrix with as many diagonal blocks as possible: $$ A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 3 \end{array}\right], \quad B=\left[\begin{array}{lllll} 1 & 2 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 5 & 0 & 0 \\ 0 & 0 & 0 & 0 & 6 \end{array}\right], \quad C=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 2 & 0 & 0 \end{array}\right] $$

Find \(M^{2}\) and \(M^{3}\) for (a) \(M=$$\left[\begin{array}{ccccc}2 & 0 & 0 & 0 \\\ 0 & , 1 & 4 & & 0 \\ 0 & -1 & 2 & -\frac{1}{1} & -0 \\ 0 & 10 & 0 & 3\end{array}\right]\) (b) \(M=\) \(\left[\begin{array}{cccc}1 & 1 & 0 & 0 \\ 2 & 3 & 0 & 0 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 14 & 5\end{array}\right]\)

Suppose \(U=\left[U_{i k}\right]\) and \(V=\left[V_{k j}\right]\) are block matrices for which \(U V\) is defined and the number of columns of each block \(U_{i k}\) is equal to the number of rows of each block \(V_{k j}\). Show that \(U V=\left[W_{i j}\right]\), where \(W_{i j}=\sum_{k} U_{i k} V_{k j}\)

Suppose \(M\) and \(N\) are block diagonal matrices where corresponding blocks have the same size, say \(M=\operatorname{diag}\left(A_{i}\right)\) and \(N=\operatorname{diag}\left(B_{i}\right)\). Show (i) \(M+N=\operatorname{diag}\left(A_{i}+B_{i}\right)\), (iii) \(\quad M N=\operatorname{diag}\left(A_{i} B_{i}\right)\), (ii) \(k M=\operatorname{diag}\left(k A_{i}\right)\), (iv) \(f(M)=\operatorname{diag}\left(f\left(A_{i}\right)\right)\) for any polynomial \(f(x)\).