Matrices

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

\[\boldsymbol{A}\,(\boldsymbol{B}+\boldsymbol{C})\ =\ \boldsymbol{A}\boldsymbol{B}+\boldsymbol{A}\boldsymbol{C}\,.\]

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

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

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

Is \(\ \,\text{tr}\,(\boldsymbol{A}\boldsymbol{B})= \text{tr}\boldsymbol{A}\cdot\text{tr}\boldsymbol{B}\ \) ?

Exercise 2. \(\,\) Observe on the example of matrices

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

that the identity

(1)\[(\boldsymbol{A}+\boldsymbol{B})^2\ =\ \boldsymbol{A}^2+2\boldsymbol{A}\boldsymbol{B}+\boldsymbol{B}^2\]

does not hold in a matrix algebra.

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

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

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

  1. Compute \(\ \,\boldsymbol{P}\boldsymbol{Q},\ \,\boldsymbol{Q}\boldsymbol{P},\ \boldsymbol{P}^2,\ \boldsymbol{Q}^2.\)

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

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

Hint to point 3. \(\ \boldsymbol{P}\ \) and \(\ \boldsymbol{Q}\ \) 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. \(\,\) Experiment with small exponents \(\ n=2,3,4,\,\ldots\) \(\\\) to find a formula for the \(\ n\)-th power of the following matrices over the field \(\ Q:\)

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

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

Exercise 5. \(\,\) Let

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

Compute \(\ \boldsymbol{A}^n\ \) and \(\ \boldsymbol{A}^n\boldsymbol{v}\ \) for arbitrary \(\ n\in\boldsymbol{N}\ \).

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

(2)\[\begin{split}\boldsymbol{A}\boldsymbol{X}_1\ =\ \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right]\,,\qquad \boldsymbol{A}\boldsymbol{X}_2\ =\ \left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right]\,,\qquad \boldsymbol{A}\boldsymbol{X}_3\ =\ \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right]\end{split}\]

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

  1. What is the effect of matrix multiplication \(\ \boldsymbol{A}\boldsymbol{X}\,\) ?

  2. Assume that \(\ \boldsymbol{A}\ \) is a nondegenerate square matrix of order 3.: \(\ \det\boldsymbol{A}\ne0.\) What is the meaning of the matrix \(\ \boldsymbol{X}\ \) ?

  3. Assume that \(\ \boldsymbol{A}\ \) is a degenerate square matrix of order 3.: \(\ \det\boldsymbol{A} = 0.\)

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

Hints.

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

  1. 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 \(\ \boldsymbol{R}_1,\boldsymbol{R}_2,\boldsymbol{R}_3\ \) the consecutive rows of the matrix \(\ \boldsymbol{A}.\) \(\\\) Since \(\ \det\boldsymbol{A}=0,\ \) then rank of the matrix \(\ \boldsymbol{A}\ \) equals 1 or 2.

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

    If \(\ \,\boldsymbol{R}_1\boldsymbol{X}=1,\ \,\) then \(\ \,\boldsymbol{R}_2\boldsymbol{X}=c_2\,(\boldsymbol{R}_1\boldsymbol{X})=c_2\ \,\) and \(\ \,\boldsymbol{R}_3\boldsymbol{X}=c_3\,(\boldsymbol{R}_1\boldsymbol{X})=c_3.\) \(\\\) 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. \(\ \text{rk}\boldsymbol{A}=2.\ \ \) Without loss of generality (up to the order of the rows) we may assume that \(\ \,\boldsymbol{A}= \left[\begin{array}{c} \boldsymbol{R}_1 \\ \boldsymbol{R}_2 \\ c_1\boldsymbol{R}_1+c_2\boldsymbol{R}_2 \end{array}\right],\)
    where \(\ \boldsymbol{R}_1,\,\boldsymbol{R}_2\ \) 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:

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

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 \(\ n=2,3,\,\ldots\)

Exercise 8. \(\,\) Find the inverse of the matrix

\[\begin{split}\boldsymbol{A}\ =\ \left[\begin{array}{rrrrr} 1 & -1 & 1 & -1 & 1 \\ 0 & 1 & -1 & 1 & -1 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right]\,.\end{split}\]

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 \(\ \boldsymbol{X}\ \) from an equation:

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

Exercise 10. \(\,\) Solve a matrix equation:

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

Exercise 11. \(\\\) 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 \(\ \,\boldsymbol{A}\,=\, \left[\begin{array}{rr} 1 & 2 \\ 1 & 1 \end{array}\right]\,\in M_2(R),\ \) i.e., find all the matrices \(\ \boldsymbol{X}\in M_2(R),\ \) such that \(\ \,\boldsymbol{A}\boldsymbol{X}=\boldsymbol{X}\boldsymbol{A}.\) \(\\\) 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.


© Copyright 2018, Jan Aksamit, Joanna Marzec, and Marcin Kostur. Last updated on Jan 29, 2022. Created using Sphinx 4.0.1.