Posted by Jason Polak on 02. April 2013 · Write a comment · Categories: elementary · Tags: , , ,

I sometimes am a teaching assistant for MATH 133 at McGill, and introductory linear algebra course that covers linear systems, diagonalisation, geometry in two and three dimensional Euclidean space, and the vector space $ \mathbb{R}^n$, and I’ve collected a few theoretical questions here that I like to use in the hope that they may be useful to people studying this kind of material. I made up all of these questions, although obviously many of them in form are the same as elsewhere. Some of the questions are a bit unusual and curious, and none of them need special tricks to solve, just an understanding of the concepts in the course. They focus mostly on understanding the theory, and there are very few straight computational-type questions here.

Note. None of these questions are official in the sense that I do not write the final exams. The exact syllabus of the course should always be taken to be the class material and official course material on the course website. These are more for extra practice, but do cover the material of the course fairly closely.

Conventions. Matrices are matrices over the real numbers, and vector spaces are real vector spaces. Actually, all vector spaces considered will be $ \mathbb{R}^n$ for some $ n$ and subspaces since the class does not cover abstract vector spaces.

    Linear Systems

  1. Homogeneous Systems
  2. Suppose the homogenous system $ AX = 0$ has a unique solution and the rank of $ A$ is $ 5$. What are the dimensions of $ A$?

  3. Homogenous Systems II
  4. True or false: Suppose the system $ AX = B$ has infinitely many solutions. Then $ AX = 0$ has infinitely many solutions.

  5. Homogeneous Systems III
  6. True or false: if $ AX = 0$ has infinitely many solutions then $ A^tX = 0$ has infinitely many solutions.

  7. Parameters
  8. Let $ A$ be a $ 3\times 3$ matrix and suppose it has two leading $ 1$’s (i.e. it has rank two). Which of the following must be true?

    1. The system $ AX = B$ has infinitely many solutions.
    2. The system $ AX = B$ has at least one solution.
    3. The system $ AX = 0$ has infinitely many solutions.
    4. The system $ AX = 0$ has at least one solution.
    5. The system $ AX = B$ has no solutions.
  9. Different Kinds of Parameters
  10. For which values of $ a$ and $ b$ does the system

    $ 4x + 6y = 10\\2x + ay = b$

    have infinitely many solutions? Are there any values of $ a$ and $ b$ such that the system has \emph{no} solutions?

  11. Different Kinds of Parameters II
  12. For which values of $ a$ and $ b$ does the system

    $ 2x + 10y = m\\ax + by = 0$

    have a solution for all possible values of $ m$?

    Matrices

  13. Zero or Nonzero?
  14. For which numbers $ n=1,2,3,4,\dots$ is the following statement true: if $ A$ and $ B$ are any $ n\times n$ matrices such that $ AB = 0$ then $ BA = 0$.

  15. Well-Defined
  16. Suppose $ A$ is a $ 3\times 2$ matrix, $ B$ is a $ 2\times 3$ matrix and $ C$ is a $ 3\times 8$ matrix. Which of the following expressions are well-defined?

    1. $ AB – C$
    2. $ AB – BA$
    3. $ ABC$
    4. $ CBA$
    5. $ BAC$
    6. $ A^tC$
  17. Real or Fake
  18. Suppose that $ A$ and $ J$ are $ 2\times 2$ matrices such that $ AJ = A$. Must $ J$ be the identity matrix?

  19. Real or Fake II
  20. Suppose that $ J$ is an $ n\times n$ matrix such that $ AJ = A$ for every $ n\times n$ matrix $ A$. Must $ J$ be the identity matrix?

  21. Symmetric Matrices
  22. A matrix $ A$ is called symmetric if $ A = A^t$. What can you say about the dimensions of $ A$?

  23. Symmetric Matrices II
  24. If $ A$ is an $ n\times n$ matrix, show that $ AA^t$ is a symmetric matrix.

  25. Symmetric Matrices III**
  26. Suppose that $ B$ is an $ n\times n$ symmetric matrix. Can we conclude that there is some matrix $ A$ such that $ B = AA^t$?

  27. Commuting Matrices
  28. Suppose that $ A$ is a $ 2\times 2$ matrix and $ AB = BA$ for all $ 2\times 2$ matrices $ B$. Show that $ A$ is a constant multiple of the identity matrix.

    Determinants

  29. Determinants and Products
  30. If $ A$ and $ B$ are $ n\times n$ matrices, is is true that $ \det(AB) = \det(BA)$? Try some small matrices first!

  31. Invertibility?
  32. Suppose $ A,B,C,$ and $ D$ are $ n\times n$ matrices and $ AB^2C^tABC^5D^9AD^t$ is invertible. Can we conclude that $ A$ is invertible?

  33. Inverses Inverses
  34. Suppose that $ A$ is an invertible matrix. Is it true that $ (A^{-1})^{-1} = A$ (in words: the inverse of the inverse of $ A$ is $ A$)? Explain.

  35. Inverses and Solutions
  36. Suppose $ A$ is an $ n\times n$ matrix that is not invertible. Which of the following statements are true?

    1. The homogenous system $ AX = 0$ has infinitely many solutions.
    2. The system $ AX = B$ has infinitely many solutions for any column $ B$.
  37. Elementary Matrices
  38. Find a matrix that cannot be written as the product of elementary matrices, and justify your answer.

  39. Computing Determinants
  40. Suppose $ A$ is a $ 3\times 3$ matrix such that $ \det(A) = 4$ and $ B$ is a $ 3\times 3$ matrix with $ \det(B) = -3$. Compute the following determinants:

    1. $ \det(A^2B)$
    2. $ \det(A^tB)$
    3. $ \det(AB^4)$
    4. $ \det(\mathrm{adj}(A)B)$
    5. $ \det(\mathrm{adj}(AB))$
    6. $ \det(5\mathrm{adj}(-6B))$
    7. $ \det(3A^t(A^tB^4))$
  41. Computing Determinants II**
  42. Suppose $ A$ is an $ n\times n$ matrix. Find the determinant of

    $ \mathrm{adj}(\det[\mathrm{adj}(\mathrm{adj}(\mathrm{adj}(A)))]A)$

    in terms of $ \det(A)$ and $ n$. Hint: your formula should have an $ n$ and $ \det(A)$ in it. First find a formula for $ \det(\mathrm{adj}(A))$ in terms of $ n$ and $ \det(A)$ and then apply this formula multiple times.

    Diagonalization

  43. Characteristic Polynomial
  44. Suppose $ A$ and $ B$ are square matrices with characteristic polynomials $ p(x)$ and $ q(x)$ respectively. True or false: $ AB$ has characteristic polynomial $ p(x)q(x)$.

  45. Characteristic Polynomial II
  46. Suppose a matrix $ A$ has characteristic polynomial

    $ p(x) = x^2 + 5x + 6.$

    Is $ A$ necessarily diagonalizable?

  47. Characteristic Polynomial III
  48. Suppose a matrix $ A$ has characteristic polynomial

    $ p(x) = x.$

    Is $ A$ necessarily diagonalizable?

  49. Rank and Diagonalization
  50. A square matrix $ A$ has characteristic polynomial

    $ p(x) = (x-3)^4(x-2)^8.$

    Suppose $ A$ is diagonalizable. Determine the rank of the following matrices:

    1. $ A – 3I$
    2. $ A + 3I$
    3. $ A – 2I$
    4. $ A + 2I$
    5. $ A – 10I$
  51. Rank and Diagonalization II
  52. Suppose that a square matrix $ A$ has characteristic polynomial

    $ p(x) = (x-1)(x + 1)(x-2)(x + 2)(x-3)(x + 3)(x-200)^2.$

    Suppose further than $ A$ is not diagonalizable. Determine the rank of the following matrices:

    1. $ A – I$
    2. $ A + 3I$
    3. $ A – 200I$
  53. Lonely Eigenvalue
  54. Let $ A$ be an $ n\times n$ matrix with \emph{exactly one} eigenvalue $ \lambda$. Find the one true statement:

    1. The matrix $ A + \lambda I_n$ can be reduced to the identity using elementary row operations.
    2. The matrix $ A – \lambda I_n$ is invertible.
    3. If $ A$ is diagonalizable, then $ A – \lambda I_n$ has rank $ 1$.
  55. Parameter Dependence
  56. For which values of $ m$ is the matrix

    $ \begin{pmatrix}2 & m\\m & 2\end{pmatrix}$

    diagonalizable?

  57. When Can You Diagonalize?
  58. Which of the following statements are true for any $ n\times n$ matrix $ A$? For any false statements you find, provide a counterexample.

    1. If $ A$ has $ n$ distinct eigenvalues, then $ A$ is diagonalizable.
    2. If $ A$ has fewer than $ n$ distinct eigenvalues, then $ A$ is not diagonalizable.
    3. If $ A$ has $ 0$ as an eigenvalue, then $ A$ is not diagonalizable.
    4. If $ A$ has $ 0$ as an eigenvalue, then $ A$ is not invertible.
    5. If $ \det(A) = 0$ then $ A$ is not diagonalizable.
    6. If $ \det(A) = 0$ then $ A$ has $ 0$ as its only eigenvalue.
  59. Matrix Operations and Diagonalization
  60. Which of the following statements are true for any $ n\times n$ matrices $ A$ and $ B$? For any false statements you find, provide a counterexample.

    1. If $ A$ and $ B$ are diagonalizable, then $ A + B$ is diagonalizable.
    2. If $ A$ and $ B$ are diagonalizable, then $ AB$ is diagonalizable.
    3. If $ A$ is diagonalizable then $ A^k$ is diagonalizable.
    4. Either $ A$ or $ A + I$ is diagonalizable.
    5. $ A + kI$ is diagonalizable for some real number $ k$.
  61. Similarity
  62. Suppose $ A$ is diagonalizable and $ B = P^{-1}AP$ for some invertible matrix $ P$. Which of the following statements are necessarily true?

    1. $ B$ is diagonalizable
    2. $ P$ is diagonalizable
  63. Computation
  64. Consider the matrix

    $ \begin{pmatrix}2 & -9\\-2 & 5\end{pmatrix}.
    $

    Find a formula for $ A^n$ and use it to compute $ A^4$.

    Geometry

  65. Lengths
  66. Let $ v\in \mathbb{R}^n$. True or false:

    $ \frac{1}{\lVert v\rVert}v$

    is a vector that has length one.

  67. Distance Between a Line and a Point
  68. Find the distance in $ \mathbb{R}^2$ between the line $ y = 3x + 8$ and the point $ P(10,5)$.

  69. Distance Between a Line and a Point II**
  70. Suppose that $ L$ is a line in $ \mathbb{R}^3$, $ Q$ a point on $ L$, and $ P$ a point not on $ L$. Suppose that $ \vec{n}$ is a (nonzero) vector perpendicular to the direction vector of $ L$.

    Is $ \lVert\mathrm{proj}{\vec{n}\rVert{\vec{PQ}}}$ the distance between $ L$ and $ P$? Explain.

  71. Points, Points
  72. Suppose that $ P,Q,R,S$ are distinct points in $ \mathbb{R}^2$ such that $ \vec{PQ}$ and $ \vec{RS}$ are parallel. Is it necessarily true that $ \vec{PR}$ and $ \vec{QS}$ are parallel?

  73. Lots of Points
  74. Suppose $ P_1,\dots,P_n$ are $ n$ distinct points in $ \mathbb{R}^2$ such that $ \vec{P_kP_{k+1}}$ is parallel to $ \vec{P_{k+1}P_{k+2}}$ for $ k = 1,\dots,n-2$. Is $ \vec{P_1P_n}$ parallel to $ \vec{P_1P_2}$?

  75. Defined Things
  76. Which of the following expressions are well-defined for all $ u,v,w\in\mathbb{R}^3$? Note that $ u\times v$ denotes the cross product and $ u\cdot v$ denotes the dot product.

    1. $ u\times v$
    2. $ u\cdot v$
    3. $ u\times(u\cdot v)$
    4. $ u\cdot (u\times v)$
    5. $ \tfrac{1}{u\times v}u$
    6. $ \tfrac{1}{u\cdot v}v$
    7. $ u\times(v\times w)$
    8. $ u\cdot(v\cdot w)$
    9. $ (u\times w)\cdot (v\times w)$
    10. $ (u\cdot v)\times (v\cdot u)$
    11. $ v\times ( (u\cdot u)v)$
    12. $ v\cdot ( (u\times u)\times v)$

    Linear Transformations and The Vector Space $ \mathbb{R}^n$

  77. Examples
  78. Find the matrix of the linear transformation in $ \mathbb{R}^2$ with respect to the standard basis:

    1. Rotation by an angle of $ \theta = \tfrac{\pi}{6}$ counterclockwise.
    2. Rotation by an angle of $ \theta = \tfrac{\pi}{12}$ counterclockwise (your answer should be expressed only with roots and numbers, not with $ \sin(x)$ or $ \cos(x)$ appearing in your formula.
    3. Reflection about the line $ y = 10x$.
    4. Rotation by an angle of $ \theta = \tfrac{\pi}{4}$ clockwise followed by a reflection in the line $ y = -x$. How does this compare to the matrix of rotation counterclockwise by an angle of $ \tfrac{\pi}{4}$.
  79. Rotation, Reflection
  80. Is it possible to write the composition of a reflection and a rotation just as a rotation?

  81. Tricky Examples
  82. Which of the following are linear transformations?

    1. $ f:\mathbb{R}\to \mathbb{R}$ given by $ f(x) = 3x + 4$
    2. $ f:\mathbb{R}\to \mathbb{R}$ given by $ f(x) = |x|$
    3. $ f:\mathbb{R}\to \mathbb{R}^2$ given by $ f(x) = [4x~~~0]^t$
    4. Let $ v\in\mathbb{R}^2$ be a fixed vector. Define $ f:\mathbb{R}^2\to \mathbb{R}$ by $ f(u) = u\cdot v$ where $ u\cdot v$ is the dot product between $ u$ and $ v$.
    5. Let $ v\in \mathbb{R}^3$ be a fixed vector. Define $ f:\mathbb{R}^3\to\mathbb{R}^3$ by $ f(u) = u\times v$ where $ u\times v$ is the cross product between $ u$ and $ v$.
    6. Let $ v\in \mathbb{R}^2$ be a fixed vector. Define $ f:\mathbb{R}^2\to \mathbb{R}$ by $ \det[u~~~v]$ where $ [u~~~v]$ is the matrix with $ u$ as the first column and $ v$ as the second column (hint: do an example).
    7. Let $ v\in\mathbb{R}^2$ be a fixed vector. Define $ f:\mathbb{R}^2\to \mathbb{R}$ by $ \mathrm{trace}[u~~~v]$.
  83. Effects
  84. Suppose $ T:\mathbb{R}^3\to\mathbb{R}^2$ is a linear transformation and we had the vectors

    $ T[1~~~ 3~~~ 4]^t\\T[2~~~ 1~~~ 1]^t\\T[4~~~ 7~~~ 9]^t$

    Could we then determine $ T[x~~~y~~~z]^t$ for any $ x,y,z\in \mathbb{R}$? In other words, could we compute what $ T$ does to any vector?

  85. Spanning Matrices
  86. Suppose $ A$ and $ B$ are $ n\times n$ matrices, that the columns of $ A$ span $ \mathbb{R}^n$ and the rows of $ B$ span $ \mathbb{R}^n$. Is it necessarily true that the columns of $ AB$ span $ \mathbb{R}^n$?

  87. Spanning Matrices II
  88. Suppose $ A$ is a $ 2\times 2$ matrix:

    $ \begin{pmatrix}a & b\\c & d\end{pmatrix}$

    whose columns span $ \mathbb{R}^2$. If we put $ A$ in a bigger matrix:

    $ B = \begin{pmatrix}a & b & 0\\c & d & 0\\0 & 0 & 1\end{pmatrix}$

    Do the rows of $ B$ span $ \mathbb{R}^3$? Do the columns of $ B$ span $ \mathbb{R}^3$?

  89. Spanning Matrices III
  90. Suppose $ A$ and $ B$ are $ n\times n$ matrices, and the columns of $ A$ span $ \mathbb{R}^3$, but that the columns of $ B$ do \emph{not} span $ \mathbb{R}^3$. Can the columns of $ AB$ span $ \mathbb{R}^3$?

  91. Matrix Transformations
  92. Suppose that $ A$ is a $ 2\times 2$ matrix such that $ AX = B$ has a solution for every $ B\in\mathbb{R}^2$. Which of the following statements is true?

    1. The system $ AX = 0$ has infinitely many solutions.
    2. The matrix $ A$ cannot be reduced to the identity matrix using elementary row operations.
    3. $ \det(A) \not= 0$.
    4. The matrix $ A$ does not have $ 0$ as an eigenvalue.
  93. Matrix Transformations II
  94. Suppose $ A$ is the matrix in the standard basis of the linear transformation $ T$ that does the following: $ T([1~~~0]^t)$ is the vector obtained by rotating the vector $ [1~~~0]^t$ by an angle of $ \tfrac{\pi}{6}$ and $ T([0~~~1]^t)$ is the vector obtained by reflecting the vector $ [0~~~1]^t$ in the line $ y = -x$. Which of the following statements are true? Explain.

    1. No such linear transformation can exist, because a linear transformation cannot rotate one basis vector and reflect the other. Madness!
    2. The matrix $ A$ can be written as a product of elementary matrices.
    3. The matrix $ A$ has eigenvalue $ 0$.
    4. $ \det(A)\not = 0$.
  95. Spanning and Counting
  96. Find the number of $ 3\times 3$ matrices that are invertible and whose only entries are $ 0$ and $ 1$.

  97. Spanning and Counting II**
  98. Find the number of $ n\times n$ matrices that are invertible and whose only entries are $ 0$ and $ 1$? (Your formula should be a function of $ n$).

  99. Spanning
  100. For what values of $ m$ are the following vectors linearly independent?

    $ [1~~~2~~~3]^t,[1~~~1~~~-1]^t,[2~~~3~~~m]^t$
  101. Spanning II
  102. For what values of $ m$ are the following vectors linearly independent?

    $ [1~~~4~~~2]^t, [m~~~3~~~2]^t,[m-1~~~m+1~~~8]^t$
  103. Linear Transformation Dimensions
  104. Suppose $ A$ is a $ 5\times 5$ matrix with $ \det(A) = 0$. Which of the following statements are true?

    1. $ \mathrm{dim}(\mathrm{im}(A)) = 5$
    2. $ \mathrm{dim}(\mathrm{im}(A)) < 5$
    3. The system $ AX = 0$ has infinitely many solutions.
  105. Null space
  106. Suppose $ A$ is an $ n\times n$ matrix. Show that $ AX= 0$ has a unique solution if and only if $ \mathrm{dim}(\mathrm{im}(A)) = n$.

  107. Null Space II
  108. Suppose $ A$ and $ B$ are $ n\times n$ matrices and $ \mathrm{dim}(\mathrm{null}(AB)) = 0$. Which of the following statements are necessarily true? For the statements that are not necessarily true, provide a counterexample.

    1. $ \mathrm{dim}(\mathrm{null}(A)) = 0$
    2. $ \mathrm{dim}(\mathrm{null}(B)) = 0$
    3. $ \mathrm{dim}(\mathrm{im}(A)) = n$
    4. $ \mathrm{dim}(\mathrm{im}(B)) = n$.
  109. Image
  110. Suppose $ A,B$ and $ C$ are $ n\times n$ matrices and $ \mathrm{dim}(\mathrm{im}(ABC)) = n$. Which of the following statements are necessarily true? For the statements that are not necessarily true, provide a counterexample.

    1. $ \mathrm{dim}(\mathrm{im}(C)) = n$
    2. $ \mathrm{dim}(\mathrm{im}(B)) = n$
    3. $ \mathrm{dim}(\mathrm{im}(A)) = n$
    4. $ \mathrm{dim}(\mathrm{null}(C)) = 0$
    5. $ \mathrm{dim}(\mathrm{null}(B)) = 0$
    6. $ \mathrm{dim}(\mathrm{null}(A)) = 0$
  111. Linear Independence
  112. If $ v_1,v_2,v_3\in\mathbb{R}^n$ are linearly dependent and $ f:\mathbb{R}^n\to \mathbb{R}^n$ is a linear transformation, show that $ f(v_1),f(v_2),f(v_3)$ are linearly dependent.

  113. Linear Independence II
  114. If $ v_1,v_2,v_3\in\mathbb{R}^n$ are linearly independent, which of the following statements are true?

    1. For every linear transformation $ f:\mathbb{R}^n\to R^n$, the vectors $ f(v_1),f(v_2),f(v_3)$ are linearly independent.
    2. There exists a linear transformation $ f:\mathbb{R}^n\to\mathbb{R}^2$ such that the vectors $ f(v_1),f(v_2),f(v_3)$ are linearly independent.
    3. There exists infinitely many linear transformations $ f:\mathbb{R}^n\to\mathbb{R}^3$ such that $ f(v_1),f(v_2),f(v_3)$ are linearly independent.
    4. There is a unique linear transformation $ f:\mathbb{R}^n\to\mathbb{R}^3$ such that the vectors $ f(v_1),f(v_2),f(v_3)$ are linearly independent.
  115. Subspaces
  116. Which of the following subsets of $ \mathbb{R}^3$ are subspaces of $ \mathbb{R}^3$?

    1. Let $ c\in \mathbb{R}$ be a constant. $ \{ [m~~~n~~~0]^t : m – n = c^2\} $ .
    2. $ \{ [m~~~n~~~2n]^t : mn = 0\}$.
    3. $ \{ [m~~~0~~~m^2]^t : m^4 = 0\}$.
    4. $ \{ [m~~~0~~~0]^t : \sin(m) = 1\}$.
    5. $ \{ [m~~~n~~~p]^t : m^3 = n^3\}$.
    6. $ \{ [m~~~0~~~0]^t : \mathrm{det}(\begin{pmatrix}m & 2\\ 0 & 11\end{pmatrix}) = 0 \}$
    7. $ \{ [m~~~n~~~p]^t : 4m + 2n – 10p = 0\}$
    8. $ \{ v\in \mathbb{R}^3 : v\cdot [2~~~4~~~-9]^t = 0\}$
    9. $ \{ v\in \mathbb{R}^3 : v\cdot [1~~~2~~~50]^t \not=0\}$
    10. $ \{ v\in \mathbb{R}^3 : v\times [-9~~~-1~~~5]^t = 0\}$
    11. $ \{ v\in \mathbb{R}^3 : v\times [1~~~10~~~1]^t = 1\}$
    12. $ \{ v\in \mathbb{R}^3 : \{ v, [1~~~1~~~1]^t\}\text{~~~are linearly independent~~~}\}$
    13. $ \{ [m~~~n~~~p]^t : m^2 + n^2 + p^2 \geq 0\}$
  117. Subspaces II
  118. Which of the following subsets $ S$ of $ \mathbb{R}^n$ are necessarily subspaces of $ \mathbb{R}^n$?

    1. Let $ f:\mathbb{R}^n\to\mathbb{R}^n, g:\mathbb{R}^n\to \mathbb{R}^n$ be two linear transformations and define $ S$ to be the set of vectors $ v\in \mathbb{R}^n$ such that $ f(v), g(v)$ are linearly independent.
    2. Let $ f:\mathbb{R}^n\to\mathbb{R}^n$ and $ g:\mathbb{R}^n\to\mathbb{R}$ be two linear transformations, and define $ S$ to be the set of vectors $ v\in \mathbb{R}^n$ such that $ f(v) = g(v)f(v)$.
    3. Let $ f:\mathbb{R}^n\to\mathbb{R}^n$ and $ g:\mathbb{R}^n\to\mathbb{R}$ be two linear transformations, and define the set $ S$ to be the set of vectors $ v\in \mathbb{R}^n$ such that $ g(f(v)) = g(v)$.
    4. Let $ f:\mathbb{R}^n\to \mathbb{R}$, $ g:\mathbb{R}^n\to\mathbb{R}$, and $ h:\mathbb{R}\to\mathbb{R}^n$ be three linear transformations and define $ S$ to be the set of $ v\in \mathbb{R}^n$ such that $ h(f(v) + g(v)) = v$.
  119. Domains and Codomains
  120. Let $ v\in\mathbb{R}^3$ and suppose that $ f:\mathbb{R}^3\to \mathbb{R}^3$ is a linear transformation such that $ f(w) = 0$ for all $ w\in \mathbb{R}^3$ such that $ w\not= v$. Which of the following statements are true?

    1. $ f$ is the zero transformation
    2. With respect to any basis of $ \mathbb{R}^3$, the matrix of $ f$ has rank $ 1$.
    3. With respect to any basis of $ \mathbb{R}^3$, the matrix of $ f$ has rank at most $ 1$.
    4. With respect to any basis of $ \mathbb{R}^3$, the matrix of $ f$ has zero determinant.
    5. If $ A$ is the matrix of $ \mathbb{R}^3$ with respect to the standard basis, then the system $ AX = 0$ has a unique solution.

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