Matrices

Exercise 0. $$ For three randomly chosen matrices $\mathit{A},\mathit{B},\mathit{C}\in M_3(Q)$ verify distributivity of multiplication over addition:

$$\mathit{A}(\mathit{B}+\mathit{C})=\mathit{A}\mathit{B}+\mathit{A}\mathit{C}.$$

Exercise 1. $$ For matrices $\mathit{A}=\left[\begin{array}{rrr}5& -1& 0\\ 2& 3& 1\\ -1& 2& 2\end{array}\right],$ $\mathit{B}=\left[\begin{array}{rrr}-1& 2& 0\\ 1& 3& 2\\ -2& 5& 4\end{array}\right]\in M_3(R)$

compute $\mathit{A}\mathit{B},\mathit{B}\mathit{A},$ $[\mathit{A},\mathit{B}]=\mathit{A}\mathit{B}-\mathit{B}\mathit{A}$ and also determinants and traces of these three expressions.

Verify the equalities $det(\mathit{A}\mathit{B})=det\mathit{A}\cdot det\mathit{B}=det(\mathit{B}\mathit{A}),$ $\text{tr}(\mathit{A}\mathit{B})=\text{tr}(\mathit{B}\mathit{A}).$

Is $\text{tr}(\mathit{A}\mathit{B})=\text{tr}\mathit{A}\cdot \text{tr}\mathit{B}$ ?

Exercise 2. $$ Observe on the example of matrices

$$\begin{array}{r}\mathit{A}=\left[\begin{array}{cc}1& 2\\ 0& 0\end{array}\right],\mathit{B}=\left[\begin{array}{cc}1& 0\\ 3& 0\end{array}\right]\in M_2(Q)\end{array}$$

that the identity

(1)$$(\mathit{A}+\mathit{B}{)}^2={\mathit{A}}^2+2\mathit{A}\mathit{B}+{\mathit{B}}^2$$

does not hold in a matrix algebra.

1. What is the correct formula for a square of sum or difference $(\mathit{A}\pm \mathit{B}{)}^2$ of matrices $\mathit{A},\mathit{B}\in M_n(K)$ ?

2. Under what condition on matrices $\mathit{A},\mathit{B}\in M_n(K)$ the formula (1) is true ?

Exercise 3. $$ Let $\mathit{P}=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],\mathit{Q}=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\in M_3(R):$

1. Compute $\mathit{P}\mathit{Q},\mathit{Q}\mathit{P},{\mathit{P}}^2,{\mathit{Q}}^2.$

2. What is the effect of multiplication of an arbitrary matrix $\mathit{A}\in M_3(R)$ on the left or on the right by $\mathit{P}$ or $\mathit{Q}$ ?

3. Give examples of other matrices of order three whose square is equal to the identity matrix.

Hint to point 3. $\mathit{P}$ and $\mathit{Q}$ are permutation matrices. The square of a permutation matrix is the identity matrix if and only if the square of the corresponding permutation is the identity permutation. Such a property holds for example for transpositions.

Exercise 4. $$ Experiment with small exponents $n=2,3,4,\dots$ $$ to find a formula for the $n$-th power of the following matrices over the field $Q:$

$$\begin{array}{r}\left[\begin{array}{cc}1& c\\ 0& 1\end{array}\right],\left[\begin{array}{cc}2& 2\\ 0& 0\end{array}\right],\left[\begin{array}{cc}2& 1\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right],\left[\begin{array}{cc}a& b\\ 0& 0\end{array}\right],\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right].\end{array}$$

Hint. $$ In the last case it may be helpful to use Wikipedia page on Fibonacci numbers.

Exercise 5. $$ Let

$$\begin{array}{r}\mathit{A}=\left[\begin{array}{cccc}0& 2& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 2\\ 0& 0& 0& 0\end{array}\right]\in M_4(Q),\mathit{v}=\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right]\in Q^4.\end{array}$$

Compute ${\mathit{A}}^n$ and ${\mathit{A}}^n\mathit{v}$ for arbitrary $n\in \mathit{N}$.

Exercise 6. $$ Given a rectangular matrix $\mathit{A},$ assume that the solutions ${\mathit{X}}_1,{\mathit{X}}_2,{\mathit{X}}_3$ to linear problems

(2)$$\begin{array}{r}\mathit{A}{\mathit{X}}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\mathit{A}{\mathit{X}}_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],\mathit{A}{\mathit{X}}_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\end{array}$$

are known. Consider a matrix $\mathit{X}$ whose columns are ${\mathit{X}}_1,{\mathit{X}}_2,{\mathit{X}}_3:$ $\mathit{X}=[{\mathit{X}}_1|{\mathit{X}}_2|{\mathit{X}}_3].$

1. What is the effect of matrix multiplication $\mathit{A}\mathit{X}$ ?

2. Assume that $\mathit{A}$ is a nondegenerate square matrix of order 3.: $det\mathit{A}\ne 0.$ What is the meaning of the matrix $\mathit{X}$ ?

3. Assume that $\mathit{A}$ is a degenerate square matrix of order 3.: $det\mathit{A}=0.$

   a). Explain why in such case one of the problems (2) does not have a solution.

   b). Denote by $N$ the number of linear problems without solutions. In what situations $N=1,2,3$? Give an example of each such situation.

4. Consider matrix equation $\mathit{A}\mathit{X}=\mathit{B},$ where $\mathit{A}\in M_{p\times q}(K),$ $\mathit{B}=[{\mathit{B}}_1|{\mathit{B}}_2|\dots |{\mathit{B}}_r]\in M_{p\times r}(K),$ and $\mathit{X}=[{\mathit{X}}_1|{\mathit{X}}_2|\dots |{\mathit{X}}_r]\in M_{q\times r}(K)$ is the unknown matrix. Explain why solving this equation is equivalent to solving $r$ linear systems of equations of the form $\mathit{A}{\mathit{X}}_j={\mathit{B}}_j,j=1,2,\dots ,r$ with $q$ unknowns.

Hints.

1. Apply the column rule of matrix multiplication (section Matrix Multiplication (Product of Two Matrices)).

3. Use a necessary and sufficient condition for a matrix to be invertible (section Calculation of the Inverse of a Matrix, Theorem 7).

Discussion of the point 3. $$ Denote by ${\mathit{R}}_1,{\mathit{R}}_2,{\mathit{R}}_3$ the consecutive rows of the matrix $\mathit{A}.$ $$ Since $det\mathit{A}=0,$ then rank of the matrix $\mathit{A}$ equals 1 or 2.

1. $\text{rk}\mathit{A}=1.$ Without loss of generality we may assume that $\mathit{A}=\left[\begin{array}{c}{\mathit{R}}_1\\ c_2{\mathit{R}}_1\\ c_3{\mathit{R}}_1\end{array}\right],{\mathit{R}}_1\ne \mathbf{0}$ (the other cases can be solved analogously). It is easy to see that then the second and the third problem in (2) are inconsistent. Indeed, $$ if ${\mathit{R}}_1\mathit{X}=0,$ then ${\mathit{R}}_2\mathit{X}=c_2({\mathit{R}}_1\mathit{X})=0$ and ${\mathit{R}}_3\mathit{X}=c_3({\mathit{R}}_1\mathit{X})=0,$ $$ which means that $\mathit{A}\mathit{X}=\mathbf{0}.$

   If ${\mathit{R}}_1\mathit{X}=1,$ then ${\mathit{R}}_2\mathit{X}=c_2({\mathit{R}}_1\mathit{X})=c_2$ and ${\mathit{R}}_3\mathit{X}=c_3({\mathit{R}}_1\mathit{X})=c_3.$ $$ This means that the first problem is consistent if and only if $c_2=c_3=0.$

   Hence, $$ if $c_2\ne 0$ or $c_3\ne 0,$ then $N=3,$ and if $c_2=c_3=0,$ then $N=2.$

2. $\text{rk}\mathit{A}=2.$ Without loss of generality (up to the order of the rows) we may assume that $\mathit{A}=\left[\begin{array}{c}{\mathit{R}}_1\\ {\mathit{R}}_2\\ c_1{\mathit{R}}_1+c_2{\mathit{R}}_2\end{array}\right],$

   where ${\mathit{R}}_1,{\mathit{R}}_2$ are linearly independent.

   Now the third problem of (2) is inconsistent for any values of constants $c_1,c_2.$

   Furthermore, by the same argument as above, we find that

   if $c_1\ne 0$ and $c_2\ne 0,$ then $N=3;$

   if $c_1=0,c_2\ne 0$ or $c_1\ne 0,c_2=0,$ then $N=2;$

   if $c_1=c_2=0,$ then $N=1$.

Exercise 7. $$ Find inverse matrices to:

$$\begin{array}{r}\mathit{A}=\left[\begin{array}{rrrr}1& -a& 0& 0\\ 0& 1& -b& 0\\ 0& 0& 1& -c\\ 0& 0& 0& 1\end{array}\right],{\mathit{L}}_5=\left[\begin{array}{rrrrr}1& 0& 0& 0& 0\\ 1& 1& 0& 0& 0\\ 1& 2& 1& 0& 0\\ 1& 3& 3& 1& 0\\ 1& 4& 6& 4& 1\end{array}\right].\end{array}$$

The matrix $L_5$ is a lower triangular Pascal matrix: its $k$-th row contains coefficients of the Newton’s binomial formula for $(a+b{)}^k,$ $k=0,1,2,3,4$ and the supplementary zeros.

Write the code which generates matrix $L_n$ and its inverse $L_n^{-1}$ for arbitrary order $n=2,3,\dots$

Exercise 8. $$ Find the inverse of the matrix

$$\begin{array}{r}\mathit{A}=\left[\begin{array}{rrrrr}1& -1& 1& -1& 1\\ 0& 1& -1& 1& -1\\ 0& 0& 1& -1& 1\\ 0& 0& 0& 1& -1\\ 0& 0& 0& 0& 1\end{array}\right].\end{array}$$

Generalise the answer to similar matrices of arbitary orders.

In Sage such an upper triangular matrix of order $n$ may be constructed as follows:

sage: n = 5
sage: A = matrix([[(-1)^(i+j) if j>=i else 0 for j in range(n)]
                                             for i in range(n)])

Exercise 9. $$ Determine the matrix $\mathit{X}$ from an equation:

a.) $\mathit{X}\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 1\end{array}\right]=\left[\begin{array}{ccc}6& 9& 8\\ 0& 1& 6\end{array}\right];$ b.) $\left[\begin{array}{rr}3& -1\\ 5& -2\end{array}\right]\mathit{X}\left[\begin{array}{rr}5& 6\\ 7& 8\end{array}\right]=\left[\begin{array}{rr}14& 16\\ 9& 10\end{array}\right].$

Exercise 10. $$ Solve a matrix equation:

a.) $\left[\begin{array}{rr}2& -3\\ 4& -6\end{array}\right]\mathit{X}=\left[\begin{array}{rr}1& 4\\ 2& 8\end{array}\right];$ b.) $\left[\begin{array}{cc}2& 1\\ 2& 1\end{array}\right]\mathit{X}=\left[\begin{array}{rr}1& 1\\ 1& 1\end{array}\right].$

Exercise 11. $$ Can a square matrix of order 4. whose rows consist of numbers 0, 1, 2, 3 in a certain order be invertible ? What would be the answer if the rows consisted of numbers 0, 1, 2, -3 ?

Exercise 12. $$ Find all matrices which commute with the matrix $\mathit{A}=\left[\begin{array}{rr}1& 2\\ 1& 1\end{array}\right]\in M_2(R),$ i.e., find all the matrices $\mathit{X}\in M_2(R),$ such that $\mathit{A}\mathit{X}=\mathit{X}\mathit{A}.$ $$ Observe that the solutions comprise a subalgebra of the matrix algebra $M_2(R).$ $$ Determine dimension of this subalgebra and give an example of its basis.