Exercise 11.3#
Question 1#
Consider the matrix:
\[\begin{split}
\begin{pmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12\\
13 & 14 & 15 & 16
\end{pmatrix}
\end{split}\]
Enter this matrix into your script and preform the following tasks:
Extract the second column.
Extract the central \(2\times2\) block.
Find the transpose of the matrix.
Question 2#
Consider the matrix equation: \(\mathbf{A}\mathbf{x} = \mathbf{B}\), where \(\mathbf{A}\) and \(\mathbf{B}\) are known matrices and \(\mathbf{x}\) is an unknown matrix. Write a program which solves this equation for \(\mathbf{x}\), assuming \(A\) is a square matrix. Test it with the following example:
\[\begin{split}
A = \begin{pmatrix}
2 & 6\\
8 & 3\\
\end{pmatrix}
\text{ and }
B = \begin{pmatrix}
7\\
3
\end{pmatrix}
\text{ which gives }
x = \begin{pmatrix}
-0.0714 \\
1.1905
\end{pmatrix}
\end{split}\]
Bonus: Can you write a program which solves this for non-square matrices? (see numpy.linalg.lstsq).
Test it on the following example:
\[\begin{split}
A = \begin{pmatrix}
5.3 & 4.2 & 2.9\\
3.4 & 2.7 & 12.1
\end{pmatrix}
\text{ and }
B = \begin{pmatrix}
7.4 \\
1.7
\end{pmatrix}
\text{ which gives }
x = \begin{pmatrix}
0.9580\\
0.7587\\
-0.2980
\end{pmatrix}
\end{split}\]