M3 - Invertible matrices

Example 1 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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.