Math Problems Vault

Problem V.2.1

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

\left\{ \begin{array}{l} x^{*}=a_{1} x+b_{1} y, \\ y^{*}=a_{2} x+b_{2} y, \end{array}\right.

and

\left\{ \begin{array}{l} x^{*}=c_{1} x+d_{1} y, \\ y^{*}=c_{2} x+d_{2} y . \end{array}\right.

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

\left\| \begin{array}{ll} c_{1} & d_{1} \\ c_{2} & d_{2} \end{array}\right\| \left\| \begin{array}{ll} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array}\right\| .

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Problem V.2.2

(a)(b)

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

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(a)

$\| 2\| \left\| \begin{array}{l}1 \\ 2 \\ 3\end{array}\right\| =\left\| \begin{array}{l}2 \\ 4 \\ 6\end{array}\right\|$

(b)

$\left\| \begin{array}{l}1 \\ 2 \\ 3\end{array}\right\| \| 2\| =\left\| \begin{array}{l}2 \\ 4 \\ 6\end{array}\right\|$ ?

Problem V.2.3

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

A B=\sum _{i=1}^{n} \mathbf{a}_{i} \mathbf{b}^{i} .

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Problem V.2.4

(a)(b)

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

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(a)

$(A+B)^{2}=A^{2}+2 A B+B^{2}$ ,

(b)

$(A+B)^{2}+(A-B)^{2}=2\left(A^{2}+B^{2}\right)$ ?

Problem V.2.5

(a)(b)

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

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(a)

Verify that the matrix

I=\left\| \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right\|

satisfies this equation. How can this be explained in terms of problem 1?

(b)

Find all solutions of this equation among real matrices of second order.

Problem V.2.6

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

A(z)=\left\| \begin{array}{rr} a & -b \\ b & a \end{array}\right\| .

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.

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Problem V.2.7

Find the inverse of the matrix

\left\| \begin{array}{lll} 1 & 0 & 0 \\ 1 & 1 & 2 \\ 1 & 1 & 3 \end{array}\right\| .

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Problem V.2.8

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

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