Linear mappings. Operations with matrices

Given the maps:

Prove that

Find the inverse matrix for $\left\Vert \begin{smallmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1\end{smallmatrix}\right\Vert$.

Verify that $\left\Vert \begin{smallmatrix} 0 & -1 \\ 1 & -1\end{smallmatrix}\right\Vert^{3}=E$.

Markov (or stochastic) matrices play a major role in applications

The linear maps $\varphi_{P}$ associated with Markov matrices are usually applied to special so-called probability column vectors

The consistency of these definitions, dictated by problems from the natural sciences, is seen from the following statements, which one should prove at least for $n=2$

Find ${ }^{t} H \cdot H$, if

Associating with a cycle of length $n$ in $S_{n}$ (see § 8, Ch. 1) the permutation matrix (of the rows of the identity matrix $E_{n}$)

verify that $P^{n}=E$.

Show that

for any two $m \times n$ matrices $A$ and $B$.

Show that for any $m \times s$ matrix $A$ and any $s \times n$ matrix $B$ the inequality

holds.

Show that if $A B C=0$ for square matrices $A, B, C$ of order $n$, then

Find the rank of the matrix

Show that if $A=\left(a_{i j}\right)$ is a nonsingular symmetric matrix $\left(a_{i j}=a_{j i}\right)$, then $A^{-1}$ is also a symmetric matrix.

Find $A^{-1}$ and $F^{-1}$ if

Verify that

In particular,

Does $A^{-1}$ exist when $a d-b c=0$?

Prove that for any matrix

the relation

holds

(in other words, $A$ is a "root" of the quadratic equation $x^{2}-(a+d) x+(a d-b c)=0)$.

For $a d-b c \neq 0$, use relation (23) to find the inverse matrix $A^{-1}$.

Prove that if $\left\Vert \begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\right\Vert^{m}=0$, then $\left\Vert \begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\right\Vert^{2}=0$.

Justify the following argument. Let an $m \times s$ matrix $X$ be partitioned by horizontal and vertical lines into blocks (or cells), so that

where $X_{i 1}, \ldots , X_{i k}$ are matrices with the same number $m_{i}$ of rows $\left(m_{1}+\ldots +m_{i}=m\right)$, and $X_{1 j}, \ldots , X_{l j}$ are matrices with the same number $s_{j}$ of columns $\left(s_{1}+\ldots s_{k}=s\right)$. If now

is an $s \times n$ matrix with blocks $Y_{i j}$ of sizes $s_{i} \times n_{j}\left(n_{1}+\ldots +n_{r}=n\right)$, then it makes sense to speak of the product $Z=X Y$, and the matrix $Z=\left(z_{i j}\right)$ can likewise be regarded as a block matrix with blocks $Z_{i j}$, computed formally by formula (7):

By hypothesis, the sizes of the matrices $X_{i \nu }, Y_{\nu j}$ are such that the product $X_{i \nu } Y_{\nu j}$ makes sense. The technique of partitioning matrices into blocks is convenient even in as simple a case as

where $A, B, E, 0 \in M_{n}(\mathbb {R})$ ($E$ is the identity matrix, and 0 is the zero matrix).

Show that multiplying the matrix

by $T=\left(t_{i j}\right) \in M_{n}(\mathbb {R})$ on the left amounts to a linear combination of the rows $X_{(1)} \ldots , X_{(n)}$, and on the right, to a linear combination of the columns $X^{(1)}, \ldots , X^{(n)}$. In particular, note that if

is an upper unitriangular matrix, then

is the matrix obtained from $X$ by a chain of elementary transformations of type (II) on the rows.