Matrices -------- **Exercise 0.** :math:`\,` For three randomly chosen matrices :math:`\ \boldsymbol{A},\boldsymbol{B},\boldsymbol{C}\in M_3(Q)\ ` verify distributivity of multiplication over addition: .. math:: \boldsymbol{A}\,(\boldsymbol{B}+\boldsymbol{C})\ =\ \boldsymbol{A}\boldsymbol{B}+\boldsymbol{A}\boldsymbol{C}\,. **Exercise 1.** :math:`\,` For matrices :math:`\ \,\boldsymbol{A}= \left[\begin{array}{rrr} 5 & -1 & 0 \\ 2 & 3 & 1 \\ -1 & 2 & 2 \end{array}\right]\,,\ ` :math:`\ \boldsymbol{B}= \left[\begin{array}{rrr} -1 & 2 & 0 \\ 1 & 3 & 2 \\ -2 & 5 & 4 \end{array}\right]\ \ \in\ M_3(R)` compute :math:`\ \,\boldsymbol{A}\boldsymbol{B},\ \,\boldsymbol{B}\boldsymbol{A},\ \,` :math:`\ [\boldsymbol{A},\boldsymbol{B}]= \boldsymbol{A}\boldsymbol{B}-\boldsymbol{B}\boldsymbol{A}\ \,` and also determinants and traces of these three expressions. Verify the equalities :math:`\ \,\det(\boldsymbol{A}\boldsymbol{B})= \det\boldsymbol{A}\cdot\det\boldsymbol{B}= \det(\boldsymbol{B}\boldsymbol{A}),\ ` :math:`\ \,\text{tr}\,(\boldsymbol{A}\boldsymbol{B})= \text{tr}\,(\boldsymbol{B}\boldsymbol{A}).` Is :math:`\ \,\text{tr}\,(\boldsymbol{A}\boldsymbol{B})= \text{tr}\boldsymbol{A}\cdot\text{tr}\boldsymbol{B}\ ` ? **Exercise 2.** :math:`\,` Observe on the example of matrices .. math:: \boldsymbol{A}\ =\ \left[\begin{array}{cc} 1 & 2 \\ 0 & 0 \end{array}\right]\,,\quad \boldsymbol{B}\ =\ \left[\begin{array}{cc} 1 & 0 \\ 3 & 0 \end{array}\right]\quad \in\ M_2(Q) that the identity .. math:: :label: sum_square2 (\boldsymbol{A}+\boldsymbol{B})^2\ =\ \boldsymbol{A}^2+2\boldsymbol{A}\boldsymbol{B}+\boldsymbol{B}^2 does not hold in a matrix algebra. 1. What is the correct formula for a square of sum or difference :math:`\ (\boldsymbol{A}\pm\boldsymbol{B})^2\ \,` of matrices :math:`\ \boldsymbol{A},\boldsymbol{B}\in M_n(K)\,` ? 2. Under what condition on matrices :math:`\ \boldsymbol{A},\boldsymbol{B}\in M_n(K)\ ` the formula :eq:`sum_square2` is true ? **Exercise 3.** :math:`\,` Let :math:`\ \ \boldsymbol{P}\ =\ \left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\, , \quad\ \boldsymbol{Q}\ =\ \left[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\quad \in\ M_3(R):` 1. Compute :math:`\ \,\boldsymbol{P}\boldsymbol{Q},\ \,\boldsymbol{Q}\boldsymbol{P},\ \boldsymbol{P}^2,\ \boldsymbol{Q}^2.` 2. What is the effect of multiplication of an arbitrary matrix :math:`\ \boldsymbol{A}\in M_3(R)\ ` on the left or on the right by :math:`\ \boldsymbol{P}\ ` or :math:`\ \boldsymbol{Q}\,` ? 3. Give examples of other matrices of order three whose square is equal to the identity matrix. **Hint** to point 3. :math:`\ \boldsymbol{P}\ ` and :math:`\ \boldsymbol{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.** :math:`\,` Experiment with small exponents :math:`\ n=2,3,4,\,\ldots` :math:`\\` to find a formula for the :math:`\ n`-th power of the following matrices over the field :math:`\ Q:` .. math:: \left[\begin{array}{cc} 1 & c \\ 0 & 1 \end{array}\right]\,,\quad \left[\begin{array}{cc} 2 & 2 \\ 0 & 0 \end{array}\right]\,,\quad \left[\begin{array}{cc} 2 & 1 \\ 0 & 1 \end{array}\right]\,,\quad \left[\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}\right]\,,\quad \left[\begin{array}{cc} a & b \\ 0 & 0 \end{array}\right]\,,\quad \left[\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}\right]\,. **Hint.** :math:`\\` In the last case it may be helpful to use Wikipedia page on Fibonacci numbers. **Exercise 5.** :math:`\,` Let .. math:: \boldsymbol{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),\qquad \boldsymbol{v}\ =\ \left[\begin{array}{c} a \\ b \\ c \\ d \end{array}\right]\,\in\,Q^{\,4}\,. Compute :math:`\ \boldsymbol{A}^n\ ` and :math:`\ \boldsymbol{A}^n\boldsymbol{v}\ ` for arbitrary :math:`\ n\in\boldsymbol{N}\ `. **Exercise 6.** :math:`\,` Given a rectangular matrix :math:`\ \boldsymbol{A}\,,` assume that the solutions :math:`\ \boldsymbol{X}_1,\boldsymbol{X}_2,\boldsymbol{X}_3\ ` to linear problems .. math:: :label: 3_set \boldsymbol{A}\boldsymbol{X}_1\ =\ \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right]\,,\qquad \boldsymbol{A}\boldsymbol{X}_2\ =\ \left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right]\,,\qquad \boldsymbol{A}\boldsymbol{X}_3\ =\ \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] are known. Consider a matrix :math:`\ \boldsymbol{X}\ ` whose columns are :math:`\ \boldsymbol{X}_1,\boldsymbol{X}_2,\boldsymbol{X}_3:\ ` :math:`\ \boldsymbol{X}\,=\,[\,\boldsymbol{X}_1\,|\,\boldsymbol{X}_2\,|\,\boldsymbol{X}_3\,].\ ` 1. What is the effect of matrix multiplication :math:`\ \boldsymbol{A}\boldsymbol{X}\,` ? 2. Assume that :math:`\ \boldsymbol{A}\ ` is a nondegenerate square matrix of order 3.: :math:`\ \det\boldsymbol{A}\ne0.` What is the meaning of the matrix :math:`\ \boldsymbol{X}\ ` ? 3. Assume that :math:`\ \boldsymbol{A}\ ` is a degenerate square matrix of order 3.: :math:`\ \det\boldsymbol{A} = 0.` a). Explain why in such case one of the problems :eq:`3_set` does not have a solution. b). Denote by :math:`\ N\ ` the number of linear problems without solutions. In what situations :math:`\ N=1,\,2,\,3\ `? Give an example of each such situation. 4. Consider matrix equation :math:`\ \boldsymbol{A}\boldsymbol{X}=\boldsymbol{B},\ ` where :math:`\ \boldsymbol{A}\in M_{p\times q}(K),\ ` :math:`\ \boldsymbol{B}\,=\,[\,\boldsymbol{B}_1\,|\,\boldsymbol{B}_2\,|\,\ldots\,|\, \boldsymbol{B}_r\,]\in M_{p\times r}(K),\ ` and :math:`\ \boldsymbol{X}\,=\,[\,\boldsymbol{X}_1\,|\,\boldsymbol{X}_2\, |\,\ldots\,|\,\boldsymbol{X}_r\,]\in M_{q\times r}(K)\ ` is the unknown matrix. Explain why solving this equation is equivalent to solving :math:`\ r\ ` linear systems of equations of the form :math:`\ \boldsymbol{A}\boldsymbol{X}_j=\boldsymbol{B}_j\,,\ \ j=1,2,\ldots,r\ ` with :math:`\ q\ ` unknowns. **Hints.** 1. Apply the column rule of matrix multiplication (section :ref:`matrix_mult`). 3. Use a necessary and sufficient condition for a matrix to be invertible (section :ref:`calc_inv_matrix`, Theorem 7). **Discussion** of the point 3. :math:`\,` Denote by :math:`\ \boldsymbol{R}_1,\boldsymbol{R}_2,\boldsymbol{R}_3\ ` the consecutive rows of the matrix :math:`\ \boldsymbol{A}.` :math:`\\` Since :math:`\ \det\boldsymbol{A}=0,\ ` then rank of the matrix :math:`\ \boldsymbol{A}\ ` equals 1 or 2. 1. :math:`\ \text{rk}\boldsymbol{A}=1.\ \ ` Without loss of generality we may assume that :math:`\ \,\boldsymbol{A}= \left[\begin{array}{c} \boldsymbol{R}_1 \\ c_2\,\boldsymbol{R}_1 \\ c_3\,\boldsymbol{R}_1 \end{array}\right],\ \,\boldsymbol{R}_1\ne\boldsymbol{0}` (the other cases can be solved analogously). It is easy to see that then the second and the third problem in :eq:`3_set` are inconsistent. Indeed, :math:`\\` if :math:`\ \,\boldsymbol{R}_1\boldsymbol{X}=0,\ \,` then :math:`\ \,\boldsymbol{R}_2\boldsymbol{X}=c_2\,(\boldsymbol{R}_1\boldsymbol{X})=0\ \,` and :math:`\ \,\boldsymbol{R}_3\boldsymbol{X}=c_3\,(\boldsymbol{R}_1\boldsymbol{X})=0,` :math:`\\` which means that :math:`\ \,\boldsymbol{A}\boldsymbol{X}=\boldsymbol{0}.` If :math:`\ \,\boldsymbol{R}_1\boldsymbol{X}=1,\ \,` then :math:`\ \,\boldsymbol{R}_2\boldsymbol{X}=c_2\,(\boldsymbol{R}_1\boldsymbol{X})=c_2\ \,` and :math:`\ \,\boldsymbol{R}_3\boldsymbol{X}=c_3\,(\boldsymbol{R}_1\boldsymbol{X})=c_3.` :math:`\\` This means that the first problem is consistent if and only if :math:`\ c_2=c_3=0.` Hence, :math:`\,` if :math:`\ c_2\ne 0\ ` or :math:`\ c_3\ne 0,\ ` then :math:`\ N=3,\ \,` and if :math:`\ c_2=c_3=0,\ ` then :math:`\ N=2.` 2. | :math:`\ \text{rk}\boldsymbol{A}=2.\ \ ` Without loss of generality (up to the order of the rows) we may assume that :math:`\ \,\boldsymbol{A}= \left[\begin{array}{c} \boldsymbol{R}_1 \\ \boldsymbol{R}_2 \\ c_1\boldsymbol{R}_1+c_2\boldsymbol{R}_2 \end{array}\right],` | where :math:`\ \boldsymbol{R}_1,\,\boldsymbol{R}_2\ ` are linearly independent. | Now the third problem of :eq:`3_set` is inconsistent for any values of constants :math:`\ c_1,\,c_2.` | Furthermore, by the same argument as above, we find that | if :math:`\ c_1\ne 0\ ` and :math:`\ c_2\ne 0,\ ` then :math:`\ N=3\,;` | if :math:`\ c_1=0,\ c_2\ne 0\ ` or :math:`\ c_1\ne 0,\ c_2=0,\ ` then :math:`\ N=2\,;` | if :math:`\ c_1=c_2=0,\ ` then :math:`\ N=1\ `. **Exercise 7.** :math:`\,` Find inverse matrices to: .. math:: \boldsymbol{A}\ =\ \left[\begin{array}{rrrr} 1 & -a & 0 & 0 \\ 0 & 1 & -b & 0 \\ 0 & 0 & 1 & -c \\ 0 & 0 & 0 & 1 \end{array}\right]\,,\qquad \boldsymbol{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]\,. The matrix :math:`L_5\ ` is a lower triangular Pascal matrix: its :math:`\ k`-th row contains coefficients of the Newton's binomial formula for :math:`\ (a+b)^k\,,\ ` :math:`\ k=0,1,2,3,4\ ` and the supplementary zeros. Write the code which generates matrix :math:`\ L_n\ ` and its inverse :math:`\ L_n^{-1}\ ` for arbitrary order :math:`\ n=2,3,\,\ldots` .. sagecellserver:: **Exercise 8.** :math:`\,` Find the inverse of the matrix .. math:: \boldsymbol{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]\,. Generalise the answer to similar matrices of arbitary orders. In Sage such an upper triangular matrix of order :math:`\ n\ ` may be constructed as follows: .. code-block:: python 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.** :math:`\,` Determine the matrix :math:`\ \boldsymbol{X}\ ` from an equation: a.) :math:`\ \ \boldsymbol{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]\,;\quad\ ` b.) :math:`\ \ \left[\begin{array}{rr} 3 & -1 \\ 5 & -2 \end{array}\right]\, \boldsymbol{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.** :math:`\,` Solve a matrix equation: a.) :math:`\ \ \left[\begin{array}{rr} 2 & -3 \\ 4 & -6 \end{array}\right]\, \boldsymbol{X}\,=\, \left[\begin{array}{rr} 1 & 4 \\ 2 & 8 \end{array}\right]\,;\qquad\ ` b.) :math:`\ \ \left[\begin{array}{cc} 2 & 1 \\ 2 & 1 \end{array}\right]\, \boldsymbol{X}\,=\, \left[\begin{array}{rr} 1 & 1 \\ 1 & 1 \end{array}\right]\,.` **Exercise 11.** :math:`\\` 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.** :math:`\,` Find all matrices which commute with the matrix :math:`\ \,\boldsymbol{A}\,=\, \left[\begin{array}{rr} 1 & 2 \\ 1 & 1 \end{array}\right]\,\in M_2(R),\ ` i.e., find all the matrices :math:`\ \boldsymbol{X}\in M_2(R),\ ` such that :math:`\ \,\boldsymbol{A}\boldsymbol{X}=\boldsymbol{X}\boldsymbol{A}.` :math:`\\` Observe that the solutions comprise a subalgebra of the matrix algebra :math:`\ M_2(R).` :math:`\\` Determine dimension of this subalgebra and give an example of its basis.