Chapter V

The matrix

is given.

The matrices

are given. Is it possible to add the matrices

The matrices

are given. Compute the matrix $2 A+3 B-C$.

With what coefficients does the matrix

expand in the matrices $A$ and $B$ and $C$ of the preceding problem?

Is it possible to expand the matrix $\left\| \begin{array}{ll}1 & 3 \\ 7 & 9\end{array}\right\|$ in the matrices

Are the rows

linearly independent?

Let the affine transformations $\mathrm{f}$ and $\mathrm{g}$ be written in some coordinate system by the formulas

and

respectively. Prove that the product $\mathrm{g} \cdot \mathrm{f}$ is written by formulas of the same kind, the matrix of coefficients being equal to

Let $\| 2\|$ be the matrix of size $1 \times 1$ with entry 2. Is it true that

Let $\mathbf{a}_{1}, \ldots , \mathbf{a}_{n}$ be the columns of a matrix $A$, and $\mathbf{b}^{1}, \ldots , \mathbf{b}^{n}$ the rows of a matrix $B$. Verify that

Is it true that for any two square matrices of one and the same order

Consider the matrix equation $X^{2}+E=O$.

Let us associate with each complex number $z=a+b i$ the matrix

Verify that the equalities $A\left(z_{1}\right)+A\left(z_{2}\right)=A\left(z_{1}+z_{2}\right)$; $A(\bar{z})=A^{T}(z) ; \quad \left|z\right|^{2}=\det A ; \quad A\left(z_{1}\right) A\left(z_{2}\right)=A\left(z_{1} z_{2}\right) ; \quad A\left(z^{-1}\right)=A^{-1}(z)$ hold.

Find the inverse of the matrix

Expand the matrix of exercise 7 into a product of elementary matrices.

The matrix

is given.

A square matrix of order $n$ has a zero submatrix of order $n-1$. Estimate the rank of the matrix.

A matrix $A$ of order $n$ contains a zero submatrix of size $m \times k$, where $m+k>n$. Prove that $A$ is singular.

Let $A$ be a matrix with entries $a_{i j}, i=1, \ldots , m$; $j=1, \ldots , n$ and $\operatorname {Rg} A=1$. Prove that there exist a nonzero column $\mathbf{a}=\left\| \alpha_{1}, \ldots , \alpha_{m}\right\|$ and a nonzero row $\mathbf{b}=\left\| \beta_{1}, \ldots , \beta_{n}\right\|$, such that $a_{i j}=\alpha_{i} \beta_{j}$ for all $i$ and $j$.

$B$ a matrix of rank $r$ there are marked $r$ linearly independent rows and $r$ linearly independent columns. Prove that at their intersection there stands a minor different from zero. Show by an example that the statement is false if the number of marked rows is less than $r$.

Prove that for any matrices $A$ and $B$ of the same size the rank of the sum is not greater than the sum of the ranks.

Let the columns of a matrix $A$ of size $m \times n$ be linearly independent and let the product $A B$ be defined. Prove that $\operatorname {Rg} A B=\operatorname {Rg} B$.

Let $A$ be a square matrix of order $n$. Express $\det \alpha A$ through $\det A$.

Let $A$ be a square matrix of order $2 n+1$, and $A^{T}=-A$. Prove that $\det A=0$.

Prove that the determinant of any triangular matrix equals the product of its diagonal entries.

Compute

One and the same number $t$ has been added to each entry of a matrix $A$. Prove that the determinant of the resulting matrix is a linear function of $t$.

Compute the determinant of order $n$ :

Two quadratic polynomials $a x^{2}+b x+c$ and $\alpha x^{2}+\beta x+\gamma$, $(a, \alpha \neq 0)$ have a common root. Prove that

Prove that the equation $X^{2}+E=O$ has no solutions among real matrices of odd order.

How many inversions are there in the permutation (5, 4, 3, 2, 1)?

Let a matrix $P$ of order $n$ have the form

where $A$ and $C$ are square submatrices of orders $k$ and $n-k$, and $O$ is a zero submatrix. By means of the complete expansion formula prove that $\det P=\det A \det C$.

Let the numbers $x_{1}, x_{2}, x_{3}$ be pairwise distinct. Prove that for any $y_{1}, y_{2}, y_{3}$ there exists a unique polynomial of degree $\leqslant 2$, whose graph passes through the points with coordinates $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)$.

Using the formulas for the entries of the inverse matrix, find the inverse of the matrix

A system of linear equations with matrix $A$ is consistent for any column of constant terms if and only if the rows of the matrix $A$ are linearly independent. Prove this

The vectors $\mathbf{a}$ and $\mathbf{b}$, $\mathbf{a} \neq \mathbf{0}$, are given. By means of the Fredholm theorem prove that the equation $[\mathbf{a}, \mathbf{x}]=\mathbf{b}$ has a solution if and only if $(\mathbf{a}, \mathbf{b})=0$.

Find a fundamental matrix for the system with matrix $\left\| \begin{array}{lll}1 & 1 & 1\end{array}\right\|$.

Let $\left\| E_{r} \mid B\right\|$ be the reduced form of the matrix of a homogeneous system of equations with $n$ unknowns. Find a fundamental matrix of the system.

Let $F$ be a fundamental matrix of a system of linear equations $A \mathbf{x}=\mathbf{o}$ with $n$ unknowns and let the rows of $A$ be linearly independent. What will be a fundamental matrix of the system

Write the general solution of the system with augmented matrix

Let a matrix $F$ of size $n \times p$ be a fundamental matrix of some system of equations. Prove that $F^{\prime }$ will be a fundamental matrix of the same system if and only if there exists a nonsingular matrix $Q$ of order $p$, such that $F^{\prime }=F Q$.

A system of three equations in two unknowns is considered. Verify that the application of the Fredholm theorem to this system is equivalent to the following (geometrically obvious) statement: the vector $\mathbf{b}$ expands in the vectors $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ if and only if it is orthogonal to every vector $\mathbf{y}$ orthogonal to these vectors.

Let the system of equations $A \mathbf{x}=\mathbf{b}$ be consistent. Prove that the equation $\mathbf{a x}=\beta$ is a consequence of this system if and only if it is a linear combination of its equations.