## M3 - Invertible matrices

#### Example 1 🔗

Explain why the matrix $$Q= \left[\begin{array}{cccc} 1 & -2 & 8 & 5 \\ 1 & -1 & 4 & 3 \\ 1 & -1 & 5 & 4 \\ 1 & -1 & 0 & -1 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & 8 & 5 \\ 1 & -1 & 4 & 3 \\ 1 & -1 & 5 & 4 \\ 1 & -1 & 0 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$Q$$ is not invertible.

#### Example 2 🔗

Explain why the matrix $$Q= \left[\begin{array}{cccc} 1 & 4 & 5 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 4 & 5 & 4 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & 5 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 4 & 5 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 4 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$Q$$ is not invertible.

#### Example 3 🔗

Explain why the matrix $$P= \left[\begin{array}{cccc} -1 & 2 & -1 & 8 \\ -1 & 1 & -2 & 8 \\ -1 & 0 & -2 & 7 \\ 1 & 4 & 6 & -6 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 2 & -1 & 8 \\ -1 & 1 & -2 & 8 \\ -1 & 0 & -2 & 7 \\ 1 & 4 & 6 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$P$$ is invertible.

#### Example 4 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} 4 & -2 & 6 & -2 \\ 3 & -5 & -6 & -5 \\ -1 & 0 & -3 & 0 \\ -3 & 5 & 6 & 5 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 4 & -2 & 6 & -2 \\ 3 & -5 & -6 & -5 \\ -1 & 0 & -3 & 0 \\ -3 & 5 & 6 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 3 & 0 \\ 0 & 1 & 3 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 5 🔗

Explain why the matrix $$Q= \left[\begin{array}{cccc} 3 & 1 & 2 & -3 \\ -1 & 0 & -2 & 3 \\ 4 & 3 & -3 & 4 \\ 0 & 1 & -5 & 8 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 3 & 1 & 2 & -3 \\ -1 & 0 & -2 & 3 \\ 4 & 3 & -3 & 4 \\ 0 & 1 & -5 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$Q$$ is not invertible.

#### Example 6 🔗

Explain why the matrix $$M= \left[\begin{array}{cccc} 0 & 1 & -1 & -4 \\ -1 & 3 & -2 & -8 \\ 0 & 0 & 1 & 4 \\ 3 & -5 & 1 & 5 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & -1 & -4 \\ -1 & 3 & -2 & -8 \\ 0 & 0 & 1 & 4 \\ 3 & -5 & 1 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$M$$ is invertible.

#### Example 7 🔗

Explain why the matrix $$C= \left[\begin{array}{cccc} 2 & 2 & 7 & -3 \\ -2 & -1 & -4 & -3 \\ 1 & 0 & 1 & 3 \\ -1 & -1 & -3 & 1 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 2 & 2 & 7 & -3 \\ -2 & -1 & -4 & -3 \\ 1 & 0 & 1 & 3 \\ -1 & -1 & -3 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$C$$ is invertible.

#### Example 8 🔗

Explain why the matrix $$Q= \left[\begin{array}{cccc} 1 & -5 & 4 & -6 \\ -1 & 6 & -5 & 7 \\ 2 & -6 & 4 & -8 \\ 1 & -1 & 0 & -2 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -5 & 4 & -6 \\ -1 & 6 & -5 & 7 \\ 2 & -6 & 4 & -8 \\ 1 & -1 & 0 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -1 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$Q$$ is not invertible.

#### Example 9 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} 0 & 0 & -2 & -6 \\ 0 & 0 & 1 & 3 \\ 1 & -4 & 2 & 6 \\ -2 & 8 & 0 & 0 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 0 & -2 & -6 \\ 0 & 0 & 1 & 3 \\ 1 & -4 & 2 & 6 \\ -2 & 8 & 0 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & -4 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 10 🔗

Explain why the matrix $$C= \left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ -3 & -2 & -2 & 5 \\ 4 & 4 & 1 & 1 \\ -1 & -1 & 2 & -6 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ -3 & -2 & -2 & 5 \\ 4 & 4 & 1 & 1 \\ -1 & -1 & 2 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$C$$ is invertible.

#### Example 11 🔗

Explain why the matrix $$C= \left[\begin{array}{cccc} -1 & 4 & 7 & -3 \\ 1 & 0 & 1 & 3 \\ 2 & -3 & -4 & 6 \\ 2 & -5 & -8 & 6 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 4 & 7 & -3 \\ 1 & 0 & 1 & 3 \\ 2 & -3 & -4 & 6 \\ 2 & -5 & -8 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 3 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$C$$ is not invertible.

#### Example 12 🔗

Explain why the matrix $$B= \left[\begin{array}{cccc} -1 & 1 & 0 & 3 \\ -2 & -5 & 0 & -8 \\ -1 & -1 & 0 & -1 \\ 0 & 2 & 0 & 4 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 1 & 0 & 3 \\ -2 & -5 & 0 & -8 \\ -1 & -1 & 0 & -1 \\ 0 & 2 & 0 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$B$$ is not invertible.

#### Example 13 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} -1 & 1 & -3 & -4 \\ 0 & 1 & 3 & 3 \\ 2 & -4 & -1 & 1 \\ 3 & -6 & -7 & -4 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 1 & -3 & -4 \\ 0 & 1 & 3 & 3 \\ 2 & -4 & -1 & 1 \\ 3 & -6 & -7 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 14 🔗

Explain why the matrix $$M= \left[\begin{array}{cccc} 1 & 1 & -3 & -4 \\ -2 & -1 & 5 & 5 \\ 2 & 0 & -4 & -2 \\ -1 & 0 & 2 & 1 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -3 & -4 \\ -2 & -1 & 5 & 5 \\ 2 & 0 & -4 & -2 \\ -1 & 0 & 2 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -1 \\ 0 & 1 & -1 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$M$$ is not invertible.

#### Example 15 🔗

Explain why the matrix $$M= \left[\begin{array}{cccc} 1 & -4 & -1 & -8 \\ 1 & -3 & 0 & -5 \\ 0 & 0 & 1 & 0 \\ -1 & 2 & -6 & 3 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -4 & -1 & -8 \\ 1 & -3 & 0 & -5 \\ 0 & 0 & 1 & 0 \\ -1 & 2 & -6 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$M$$ is invertible.

#### Example 16 🔗

Explain why the matrix $$B= \left[\begin{array}{cccc} 1 & 1 & 1 & 3 \\ 2 & 3 & 1 & 2 \\ 1 & 4 & -1 & -4 \\ -3 & -4 & -2 & -4 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 1 & 3 \\ 2 & 3 & 1 & 2 \\ 1 & 4 & -1 & -4 \\ -3 & -4 & -2 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$B$$ is invertible.

#### Example 17 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} 1 & -2 & 0 & -2 \\ -3 & 6 & 1 & 6 \\ -2 & 4 & -3 & 4 \\ 4 & -8 & 3 & -8 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & 0 & -2 \\ -3 & 6 & 1 & 6 \\ -2 & 4 & -3 & 4 \\ 4 & -8 & 3 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & -2 & 0 & -2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 18 🔗

Explain why the matrix $$P= \left[\begin{array}{cccc} -1 & 1 & 2 & -1 \\ -1 & 0 & 4 & -4 \\ 0 & 0 & 1 & -2 \\ -1 & 0 & 5 & -5 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 1 & 2 & -1 \\ -1 & 0 & 4 & -4 \\ 0 & 0 & 1 & -2 \\ -1 & 0 & 5 & -5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$P$$ is invertible.

#### Example 19 🔗

Explain why the matrix $$C= \left[\begin{array}{cccc} -1 & 0 & 3 & -4 \\ -2 & -3 & 1 & 5 \\ 1 & 1 & -1 & -1 \\ -2 & -4 & 0 & 8 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 0 & 3 & -4 \\ -2 & -3 & 1 & 5 \\ 1 & 1 & -1 & -1 \\ -2 & -4 & 0 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$C$$ is not invertible.

#### Example 20 🔗

Explain why the matrix $$B= \left[\begin{array}{cccc} 1 & -1 & 3 & 1 \\ 1 & 0 & 5 & 7 \\ -1 & 0 & -4 & -5 \\ 2 & -2 & 2 & -6 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 3 & 1 \\ 1 & 0 & 5 & 7 \\ -1 & 0 & -4 & -5 \\ 2 & -2 & 2 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$B$$ is not invertible.

#### Example 21 🔗

Explain why the matrix $$Q= \left[\begin{array}{cccc} 4 & 2 & 2 & -2 \\ 1 & 3 & 8 & -3 \\ 1 & 0 & -1 & 0 \\ -2 & -2 & -4 & 2 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 4 & 2 & 2 & -2 \\ 1 & 3 & 8 & -3 \\ 1 & 0 & -1 & 0 \\ -2 & -2 & -4 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 3 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$Q$$ is not invertible.

#### Example 22 🔗

Explain why the matrix $$P= \left[\begin{array}{cccc} 1 & 0 & 5 & 2 \\ 0 & 1 & -1 & 4 \\ -1 & 1 & -5 & 2 \\ 2 & -1 & 8 & 1 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 0 & 5 & 2 \\ 0 & 1 & -1 & 4 \\ -1 & 1 & -5 & 2 \\ 2 & -1 & 8 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$P$$ is invertible.

#### Example 23 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} 0 & -1 & -4 & 5 \\ 1 & 0 & 0 & -1 \\ 1 & -1 & -3 & 3 \\ -2 & 1 & 3 & -2 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & -1 & -4 & 5 \\ 1 & 0 & 0 & -1 \\ 1 & -1 & -3 & 3 \\ -2 & 1 & 3 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 24 🔗

Explain why the matrix $$P= \left[\begin{array}{cccc} 1 & -4 & -1 & 1 \\ -1 & 5 & 2 & -3 \\ 1 & -2 & 2 & -5 \\ 2 & -6 & 0 & -2 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -4 & -1 & 1 \\ -1 & 5 & 2 & -3 \\ 1 & -2 & 2 & -5 \\ 2 & -6 & 0 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$P$$ is not invertible.

#### Example 25 🔗

Explain why the matrix $$B= \left[\begin{array}{cccc} -5 & 3 & -7 & -3 \\ 3 & -2 & 5 & 3 \\ -3 & 2 & -4 & -2 \\ -2 & 2 & -4 & -4 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -5 & 3 & -7 & -3 \\ 3 & -2 & 5 & 3 \\ -3 & 2 & -4 & -2 \\ -2 & 2 & -4 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$B$$ is not invertible.

#### Example 26 🔗

Explain why the matrix $$Q= \left[\begin{array}{cccc} 1 & 0 & 1 & -2 \\ 5 & 1 & 0 & -6 \\ -3 & -1 & 3 & 1 \\ 4 & 1 & -3 & -2 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 0 & 1 & -2 \\ 5 & 1 & 0 & -6 \\ -3 & -1 & 3 & 1 \\ 4 & 1 & -3 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$Q$$ is not invertible.

#### Example 27 🔗

Explain why the matrix $$C= \left[\begin{array}{cccc} 1 & 1 & 0 & 3 \\ 1 & -2 & 6 & -3 \\ 1 & -2 & 6 & -3 \\ 0 & 4 & -8 & 8 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 0 & 3 \\ 1 & -2 & 6 & -3 \\ 1 & -2 & 6 & -3 \\ 0 & 4 & -8 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 2 & 1 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$C$$ is not invertible.

#### Example 28 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} -1 & 2 & 0 & -1 \\ 0 & 1 & -1 & -2 \\ 0 & 2 & -2 & -4 \\ -2 & 0 & 4 & 6 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 2 & 0 & -1 \\ 0 & 1 & -1 & -2 \\ 0 & 2 & -2 & -4 \\ -2 & 0 & 4 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -3 \\ 0 & 1 & -1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 29 🔗

Explain why the matrix $$N= \left[\begin{array}{cccc} 2 & -4 & -3 & 3 \\ -4 & 5 & 2 & -3 \\ 1 & -3 & -3 & 3 \\ -2 & 5 & 6 & -8 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 2 & -4 & -3 & 3 \\ -4 & 5 & 2 & -3 \\ 1 & -3 & -3 & 3 \\ -2 & 5 & 6 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$N$$ is invertible.

#### Example 30 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} 0 & 2 & -1 & -6 \\ -2 & -3 & 4 & 7 \\ -2 & -3 & 5 & 4 \\ -1 & 0 & 1 & 0 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 2 & -1 & -6 \\ -2 & -3 & 4 & 7 \\ -2 & -3 & 5 & 4 \\ -1 & 0 & 1 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$A$$ is invertible.

#### Example 31 🔗

Explain why the matrix $$B= \left[\begin{array}{cccc} -2 & 2 & -1 & -5 \\ -1 & 2 & 0 & 3 \\ 1 & -1 & 0 & 0 \\ -1 & 0 & 1 & 3 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -2 & 2 & -1 & -5 \\ -1 & 2 & 0 & 3 \\ 1 & -1 & 0 & 0 \\ -1 & 0 & 1 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$B$$ is invertible.

#### Example 32 🔗

Explain why the matrix $$Q= \left[\begin{array}{cccc} 1 & 1 & -3 & -2 \\ 0 & 1 & -5 & -5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 2 & 5 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -3 & -2 \\ 0 & 1 & -5 & -5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 2 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$Q$$ is invertible.

#### Example 33 🔗

Explain why the matrix $$C= \left[\begin{array}{cccc} 1 & 1 & -2 & -6 \\ 0 & 1 & 3 & 1 \\ 0 & -1 & -2 & 1 \\ 0 & 1 & 5 & 6 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -2 & -6 \\ 0 & 1 & 3 & 1 \\ 0 & -1 & -2 & 1 \\ 0 & 1 & 5 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$C$$ is invertible.

#### Example 34 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} 1 & 2 & 3 & -6 \\ 0 & 1 & 0 & 1 \\ 1 & 7 & 4 & -3 \\ -1 & -2 & -2 & 4 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 3 & -6 \\ 0 & 1 & 0 & 1 \\ 1 & 7 & 4 & -3 \\ -1 & -2 & -2 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 35 🔗

Explain why the matrix $$P= \left[\begin{array}{cccc} -2 & -1 & 3 & -5 \\ -1 & -1 & 3 & 0 \\ -1 & -1 & 4 & 1 \\ 0 & 2 & -2 & -5 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -2 & -1 & 3 & -5 \\ -1 & -1 & 3 & 0 \\ -1 & -1 & 4 & 1 \\ 0 & 2 & -2 & -5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$P$$ is invertible.

#### Example 36 🔗

Explain why the matrix $$B= \left[\begin{array}{cccc} 3 & 5 & -6 & 1 \\ -2 & -5 & 4 & 1 \\ 1 & 7 & -2 & -5 \\ 1 & 1 & -2 & 1 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 3 & 5 & -6 & 1 \\ -2 & -5 & 4 & 1 \\ 1 & 7 & -2 & -5 \\ 1 & 1 & -2 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$B$$ is not invertible.

#### Example 37 🔗

Explain why the matrix $$B= \left[\begin{array}{cccc} -1 & 4 & -6 & -2 \\ 0 & 1 & -2 & -1 \\ -2 & 6 & -8 & -2 \\ 1 & -4 & 6 & 2 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 4 & -6 & -2 \\ 0 & 1 & -2 & -1 \\ -2 & 6 & -8 & -2 \\ 1 & -4 & 6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -2 \\ 0 & 1 & -2 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$B$$ is not invertible.

#### Example 38 🔗

Explain why the matrix $$M= \left[\begin{array}{cccc} 1 & 4 & -3 & -3 \\ 2 & -3 & 0 & 8 \\ 0 & -2 & 1 & 3 \\ -1 & 3 & -1 & -4 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & -3 & -3 \\ 2 & -3 & 0 & 8 \\ 0 & -2 & 1 & 3 \\ -1 & 3 & -1 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$M$$ is invertible.

#### Example 39 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} -1 & 0 & 0 & -1 \\ 1 & -1 & -1 & -1 \\ -2 & 2 & 3 & 5 \\ 1 & -1 & -1 & -1 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 0 & 0 & -1 \\ 1 & -1 & -1 & -1 \\ -2 & 2 & 3 & 5 \\ 1 & -1 & -1 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 40 🔗

Explain why the matrix $$N= \left[\begin{array}{cccc} 1 & 1 & 0 & 2 \\ -2 & -1 & -2 & 1 \\ 2 & 2 & 1 & 0 \\ -2 & -2 & 1 & -7 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 0 & 2 \\ -2 & -1 & -2 & 1 \\ 2 & 2 & 1 & 0 \\ -2 & -2 & 1 & -7 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$N$$ is invertible.

#### Example 41 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} 1 & 2 & 1 & -5 \\ 0 & 1 & 3 & -1 \\ 1 & 1 & -1 & -4 \\ 0 & 0 & 4 & 1 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 1 & -5 \\ 0 & 1 & 3 & -1 \\ 1 & 1 & -1 & -4 \\ 0 & 0 & 4 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$A$$ is invertible.

#### Example 42 🔗

Explain why the matrix $$Q= \left[\begin{array}{cccc} 2 & 3 & 0 & -1 \\ -2 & 1 & -1 & 5 \\ 2 & -4 & 5 & -8 \\ -1 & -3 & 0 & -1 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 2 & 3 & 0 & -1 \\ -2 & 1 & -1 & 5 \\ 2 & -4 & 5 & -8 \\ -1 & -3 & 0 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$Q$$ is not invertible.

#### Example 43 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} 1 & 1 & -3 & 0 \\ -1 & 0 & 5 & 1 \\ 3 & 4 & -7 & 2 \\ -3 & -7 & 1 & 0 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -3 & 0 \\ -1 & 0 & 5 & 1 \\ 3 & 4 & -7 & 2 \\ -3 & -7 & 1 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -5 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 44 🔗

Explain why the matrix $$A= \left[\begin{array}{cccc} 1 & 2 & 3 & -5 \\ 0 & 1 & 2 & -2 \\ 0 & 4 & 8 & -8 \\ 0 & 4 & 8 & -8 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 3 & -5 \\ 0 & 1 & 2 & -2 \\ 0 & 4 & 8 & -8 \\ 0 & 4 & 8 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -1 \\ 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$A$$ is not invertible.

#### Example 45 🔗

Explain why the matrix $$P= \left[\begin{array}{cccc} -1 & -2 & 1 & -8 \\ -2 & -3 & 7 & 5 \\ -1 & -1 & 3 & 3 \\ -1 & -2 & 3 & -1 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & -2 & 1 & -8 \\ -2 & -3 & 7 & 5 \\ -1 & -1 & 3 & 3 \\ -1 & -2 & 3 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$P$$ is invertible.

#### Example 46 🔗

Explain why the matrix $$B= \left[\begin{array}{cccc} 1 & 2 & 2 & 5 \\ 0 & 1 & 2 & 5 \\ 0 & -1 & -1 & -2 \\ 2 & 2 & 2 & 6 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 2 & 5 \\ 0 & 1 & 2 & 5 \\ 0 & -1 & -1 & -2 \\ 2 & 2 & 2 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$B$$ is not invertible.

#### Example 47 🔗

Explain why the matrix $$Q= \left[\begin{array}{cccc} 0 & 1 & 7 & 8 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ -1 & -1 & 1 & 0 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & 7 & 8 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ -1 & -1 & 1 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$Q$$ is not invertible.

#### Example 48 🔗

Explain why the matrix $$C= \left[\begin{array}{cccc} 0 & 1 & -1 & 3 \\ -1 & -1 & -1 & -3 \\ 2 & 3 & 2 & 8 \\ 1 & 1 & 4 & 0 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & -1 & 3 \\ -1 & -1 & -1 & -3 \\ 2 & 3 & 2 & 8 \\ 1 & 1 & 4 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$$C$$ is not invertible.

#### Example 49 🔗

Explain why the matrix $$M= \left[\begin{array}{cccc} 4 & 0 & 7 & 4 \\ -2 & 1 & -4 & -3 \\ -1 & 0 & 0 & 5 \\ 1 & 0 & 2 & 2 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} 4 & 0 & 7 & 4 \\ -2 & 1 & -4 & -3 \\ -1 & 0 & 0 & 5 \\ 1 & 0 & 2 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

$$M$$ is invertible.

#### Example 50 🔗

Explain why the matrix $$P= \left[\begin{array}{cccc} -2 & 3 & 3 & 1 \\ 3 & -5 & -6 & -2 \\ 2 & -2 & 1 & 2 \\ 3 & -4 & -3 & 0 \end{array}\right]$$ is or is not invertible.

$\operatorname{RREF} \left[\begin{array}{cccc} -2 & 3 & 3 & 1 \\ 3 & -5 & -6 & -2 \\ 2 & -2 & 1 & 2 \\ 3 & -4 & -3 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$
$$P$$ is invertible.