Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 45E from Chapter 3.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 45E
Chapter:
Problem:
[M] Compute det ATA and detAAT for several random 4 × 5 matrices and several random 5 × 6 matrices. What can you say about ATA and AAT when A has more columns than rows?
Step-by-Step Solution
Given Information
We have to use matrix program to find det $A ^ { T } A$ and det $A A ^ { r }$ for several random $4 \times 5$ and $5 \times 6$ matrices.
Step-1: Case-I
The random matrix A \[A = \left[ {\begin{array}{*{20}{l}} 8&5&8&7&5\\ 3&7&9&3&4\\ 3&2&7&7&9\\ 7&4&9&4&1 \end{array}} \right]\] Transpose of A \[{A^T} = \left[ {\begin{array}{*{20}{l}} 8&3&3&7\\ 5&7&2&4\\ 8&9&7&9\\ 7&3&7&4\\ 5&4&9&1 \end{array}} \right]\] Determinant of $A^TA$: \[\det \left( {{A^T}A} \right) = {\rm{1}}{\rm{.6589e - 08}}\] Determinant of $AA^T$: \[\det \left( {A{A^T}} \right) = {\rm{2}}{\rm{.6481e + 06}}\]
Step-2: Case-II
The random matrix A \[A = \left[ {\begin{array}{*{20}{l}} 2&7&0&1&2\\ 9&0&9&4&5\\ 8&9&6&6&1\\ 5&6&2&1&1 \end{array}} \right]\] Transpose of A \[{A^T} = \left[ {\begin{array}{*{20}{c}} 2&9&8&5\\ 7&0&9&6\\ 0&9&6&2\\ 1&4&6&1\\ 2&5&1&1 \end{array}} \right]\] Determinant of $A^TA$: \[\det \left( {{A^T}A} \right) = {\rm{ - 2}}{\rm{.8897e - 09}}\] Determinant of $AA^T$: \[\det \left( {A{A^T}} \right) = {\rm{1}}{\rm{.0690e + 06}}\]
Step-3: Case-III
The random matrix A \[A = \left[ {\begin{array}{*{20}{l}} 4&4&9&7&3&5\\ 7&4&9&0&5&3\\ 3&3&9&0&3&3\\ 7&1&4&8&8&6\\ 2&5&7&4&6&4 \end{array}} \right]\] Transpose of A \[{A^T} = \left[ {\begin{array}{*{20}{c}} 4&7&3&7&2\\ 4&4&3&1&5\\ 9&9&9&4&7\\ 7&0&0&8&4\\ 3&5&3&8&6\\ 5&3&3&6&4 \end{array}} \right]\] Determinant of $A^TA$: \[\det \left( {{A^T}A} \right) = {\rm{1}}{\rm{.1155e - 07}}\] Determinant of $AA^T$: \[\det \left( {A{A^T}} \right) = {\rm{4}}{\rm{.1806e + 07}}\]
Step-4: Case-IV
The random matrix A \[A = \left[ {\begin{array}{*{20}{l}} 7&0&4&3&1&4\\ 0&9&5&7&4&2\\ 1&0&8&9&1&5\\ 2&3&8&3&4&7\\ 2&4&7&6&6&5 \end{array}} \right]\] Transpose of A \[{A^T} = \left[ {\begin{array}{*{20}{c}} 7&0&1&2&2\\ 0&9&0&3&4\\ 4&5&8&8&7\\ 3&7&9&3&6\\ 1&4&1&4&6\\ 4&2&5&7&5 \end{array}} \right]\] Determinant of $A^TA$: \[\det \left( {{A^T}A} \right) = {\rm{7}}{\rm{.7775e - 07}}\] Determinant of $AA^T$: \[\det \left( {A{A^T}} \right) = {\rm{1}}{\rm{.4664e + 08}}\]
Step-5: Conclusion
From the above steps, we find that determinant of $A^TA$ always tends to zero whereas determinant of $AA^T$ is very large number. Thus, $A^TA$ is not an invertible matrix. Therefore,
The equation $\left( A ^ { T } A \right) \mathbf { x } = \mathbf { 0 }$ has a non-trivial solution $\mathbf { x }$
We have to use matrix program to find det $A ^ { T } A$ and det $A A ^ { r }$ for several random $4 \times 5$ and $5 \times 6$ matrices.
Step-1: Case-I
The random matrix A \[A = \left[ {\begin{array}{*{20}{l}} 8&5&8&7&5\\ 3&7&9&3&4\\ 3&2&7&7&9\\ 7&4&9&4&1 \end{array}} \right]\] Transpose of A \[{A^T} = \left[ {\begin{array}{*{20}{l}} 8&3&3&7\\ 5&7&2&4\\ 8&9&7&9\\ 7&3&7&4\\ 5&4&9&1 \end{array}} \right]\] Determinant of $A^TA$: \[\det \left( {{A^T}A} \right) = {\rm{1}}{\rm{.6589e - 08}}\] Determinant of $AA^T$: \[\det \left( {A{A^T}} \right) = {\rm{2}}{\rm{.6481e + 06}}\]
Step-2: Case-II
The random matrix A \[A = \left[ {\begin{array}{*{20}{l}} 2&7&0&1&2\\ 9&0&9&4&5\\ 8&9&6&6&1\\ 5&6&2&1&1 \end{array}} \right]\] Transpose of A \[{A^T} = \left[ {\begin{array}{*{20}{c}} 2&9&8&5\\ 7&0&9&6\\ 0&9&6&2\\ 1&4&6&1\\ 2&5&1&1 \end{array}} \right]\] Determinant of $A^TA$: \[\det \left( {{A^T}A} \right) = {\rm{ - 2}}{\rm{.8897e - 09}}\] Determinant of $AA^T$: \[\det \left( {A{A^T}} \right) = {\rm{1}}{\rm{.0690e + 06}}\]
Step-3: Case-III
The random matrix A \[A = \left[ {\begin{array}{*{20}{l}} 4&4&9&7&3&5\\ 7&4&9&0&5&3\\ 3&3&9&0&3&3\\ 7&1&4&8&8&6\\ 2&5&7&4&6&4 \end{array}} \right]\] Transpose of A \[{A^T} = \left[ {\begin{array}{*{20}{c}} 4&7&3&7&2\\ 4&4&3&1&5\\ 9&9&9&4&7\\ 7&0&0&8&4\\ 3&5&3&8&6\\ 5&3&3&6&4 \end{array}} \right]\] Determinant of $A^TA$: \[\det \left( {{A^T}A} \right) = {\rm{1}}{\rm{.1155e - 07}}\] Determinant of $AA^T$: \[\det \left( {A{A^T}} \right) = {\rm{4}}{\rm{.1806e + 07}}\]
Step-4: Case-IV
The random matrix A \[A = \left[ {\begin{array}{*{20}{l}} 7&0&4&3&1&4\\ 0&9&5&7&4&2\\ 1&0&8&9&1&5\\ 2&3&8&3&4&7\\ 2&4&7&6&6&5 \end{array}} \right]\] Transpose of A \[{A^T} = \left[ {\begin{array}{*{20}{c}} 7&0&1&2&2\\ 0&9&0&3&4\\ 4&5&8&8&7\\ 3&7&9&3&6\\ 1&4&1&4&6\\ 4&2&5&7&5 \end{array}} \right]\] Determinant of $A^TA$: \[\det \left( {{A^T}A} \right) = {\rm{7}}{\rm{.7775e - 07}}\] Determinant of $AA^T$: \[\det \left( {A{A^T}} \right) = {\rm{1}}{\rm{.4664e + 08}}\]
Step-5: Conclusion
From the above steps, we find that determinant of $A^TA$ always tends to zero whereas determinant of $AA^T$ is very large number. Thus, $A^TA$ is not an invertible matrix. Therefore,
The equation $\left( A ^ { T } A \right) \mathbf { x } = \mathbf { 0 }$ has a non-trivial solution $\mathbf { x }$