11.2. First Matrix Homework

1. Show that \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\) Hint: Start with \((\mathbf{AB})^{-1} (\mathbf{AB}) = \mathbf{I}\) and solve for \((\mathbf{AB})^{-1}\).

2. Perform matrix multiplication.

   1. \[\begin{split}\begin{bmatrix} 9 & 10 & 3 \\ 10 & 7 & 6 \\ 2 & 1 & 10 \end{bmatrix} \begin{bmatrix} 0 & 5 & 1 \\ 1 & -2 & 3 \\ 5 & 5 & 0 \end{bmatrix}\end{split}\]
   2. \[\begin{split}\begin{bmatrix} 3 & 8 \\ 4 & 2 \end{bmatrix}\begin{bmatrix} -2 & 1 \\ 7 & 5 \end{bmatrix}\end{split}\]
   3. \[\begin{split}\begin{bmatrix} 9 & 10 & 3 \\ 10 & 7 & 6 \\ 2 & 1 & 10 \end{bmatrix} \begin{bmatrix} 5 \\ 2 \\ 1 \end{bmatrix}\end{split}\]

3. Consider the matrix:

   \[\begin{split}\bf{B} = \begin{bmatrix} -1 & -4 & -1 \\ 0 & 0 & -1 \\ -1 & 0 & -1 \end{bmatrix}\end{split}\]

   1. What is the determinant of \(\bf{B}\)?

   2. Is the matrix singular? If not, what is its inverse?

4. Use Python to show that the following matrix is an orthogonal matrix.

   \[\begin{split}\mathbf{Q} = \begin{bmatrix} -0.1309 & 0.9296 & -0.3268 & -0.1090 \\ -0.4748 & -0.2278 & -0.1810 & -0.8306 \\ 0.1142 & -0.2847 & -0.9272 & 0.2149 \\ -0.8628 & -0.0533 & 0.0265 & 0.5020 \end{bmatrix}\end{split}\]

   Here is Python code to make the array (copy and paste it).

   Q = np.array([[-0.1309, 0.9296, -0.3268, -0.1090],
                 [-0.4748, -0.2278, -0.1810, -0.8306],
                 [0.1142, -0.2847, -0.9272, 0.2149],
                 [-0.8628, -0.0533, 0.0265, 0.5020]])
   

   You might use np.round() when printing the output to avoid round-off errors.

5. What is the transpose of the following matrix?

   \[\begin{split}\mathbf{A} = \begin{bmatrix} 1 & 5 & 6 \\ 2 & 3 & 7 \\ 3 & 9 & 5 \end{bmatrix}\end{split}\]