Exercises

Exercise 5.1

(See Solution 5.1.) For which values of \(a, b \in {\bf R}\) is the following matrix invertible? In this event, what is its inverse?

\[ A = \left ( \begin{array}{ccc} a & b & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 4 \end{array} \right ). \]

Exercise 5.2

(See Solution 5.2.) Let \(A\) be a square matrix such that \(A^2 = {\mathrm {id}}\) (the identity matrix). Prove that \(\det(A) = \pm 1\).

Exercise 5.3

(See Solution 5.3.) Compute the determinant of a rotation matrix (cf. Example 4.18),

\[ A = \left ( \begin{array}{cc} \cos r & -\sin r \\ \sin r & \cos r \end{array} \right ). \]

Exercise 5.4

(See Solution 5.4.) Compute the determinant of

\[ \left ( \begin{array}{cccc} 2 & 0 & 1 & 4 \\ -1 & 3 & 0 & 2 \\ 1 & 0 & 2 & -3 \\ 0 & -2 & 5 & 1 \end{array} \right ). \]

Exercise 5.5

(See Solution 5.5.) Compute the determinant of \(\left ( \begin{array}{ccc} 3 & 0 & 0 \\ 1 & 4 & 0 \\ 2 & -3 & 5 \end{array} \right )\) in three ways:

Exercise 5.6

(See Solution 5.6.) Compute the determinants of the following matrices. You should be able to do this very quickly:

\[ \left ( \begin{array}{ccc} 1 & 5 & 8 \\ 40 & -9 & 1 \\ 0 & 0 & 0 \end{array} \right ) \ \text{and } \ \left ( \begin{array}{ccc} 1 & 5 & 8 \\ 40 & -9 & 1 \\ 1 & 5 & 8 \end{array} \right ). \]

Exercise 5.7

(See Solution 5.7.) Compute the determinants of

\[ \left ( \begin{array}{cccc} 3 & 26 & -9 & 3 \\ 0 & 3 & 1 & 28 \\ 0 & 0 & 2 & 71 \\ 0 & 0 & 0 & 3 \end{array} \right ) \ \text{and } \left ( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right ). \]

Exercise 5.8

(See Solution 5.8.) Compute the inverses of

\[ \left ( \begin{array}{cc} 10 & 9 \\ 11 & 10 \end{array} \right ) \ \text{and } \ \left ( \begin{array}{ccc} 5 & 2 & -1 \\ 0 & 0 & 1 \\ 6 & 2 & 3 \end{array} \right ). \]