Exercise 3.2.5

For these problems, we can do the Gaussian elimination on the augment matrix. If the matrix is full rank¹ then we get the inverse matrix in the augmenting part.

1.

The rank is $2$ and its inverse is $\left(\begin{array}{cc}\hfill -1\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right)$.

$$\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)⇝\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 1\hfill \end{array}\right)$$

$$⇝\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 1\hfill \end{array}\right)⇝\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right)$$

2.

The rank is $1$. So there’s no inverse matrix.

3.

The rank is $2$. So there’s no inverse matrix.

4.

The rank is $3$ and its inverse is $\left(\begin{array}{ccc}\hfill -\frac{1}{2}\hfill & \hfill 3\hfill & \hfill -1\hfill \\ \hfill \frac{3}{2}\hfill & \hfill -4\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill -2\hfill & \hfill 1\hfill \end{array}\right)$.

5.

The rank is $3$ and its inverse is $\left(\begin{array}{ccc}\hfill \frac{1}{6}\hfill & \hfill -\frac{1}{3}\hfill & \hfill \frac{1}{2}\hfill \\ \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill -\frac{1}{6}\hfill & \hfill \frac{1}{3}\hfill & \hfill \frac{1}{2}\hfill \end{array}\right)$.

6.

The rank is $2$. So there’s no inverse matrix.

7.

The rank is $4$ and its inverse is $\left(\begin{array}{cccc}\hfill -51\hfill & \hfill 15\hfill & \hfill 7\hfill & \hfill 12\hfill \\ \hfill 31\hfill & \hfill -9\hfill & \hfill -4\hfill & \hfill -7\hfill \\ \hfill -10\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill -3\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$.

8.

The rank is $3$. So there’s no inverse matrix.