Is a matrix NOT invertible when there are all zeros in a row?

Question

Given the matrix $A∈M_3(Z_9)$

$$ \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 0 \\ 0 & 3 & 6 \\ \end{pmatrix} $$

By multiplying the third row by $3$ we have:

$$ \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} $$

So the matrix is not invertible because there are all zeros in a row, am I right?

Answer 1

Allowed operations for matrix in ring $R$:

• Swap two rows
• Subtract $a$ times any row from any other row where $a \in R$
• Multiply any row by $b$ where $b \in R^*$

---

Let $A \in M_3(\Bbb Z/9\Bbb Z)$ be given by: $$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 0 \\ 0 & 3 & 6 \end{pmatrix}$$

Subtract twice the first row from the second row: $$\begin{pmatrix} 1 & 2 & 3 \\ 0 & 8 & 3 \\ 0 & 3 & 6 \end{pmatrix}$$

Negate the second row: $$\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 3 & 6 \end{pmatrix}$$

Subtract twice the second row from the first row: $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 6 \\ 0 & 3 & 6 \end{pmatrix}$$

Subtract thrice the second row from the third row: $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 6 \\ 0 & 0 & 6 \end{pmatrix}$$

Subtract the third row from the second row: $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 6 \end{pmatrix}$$

Since $6$ is not invertible, $A$ is not invertible.

---

Or you could have calculated its determinant as suggested in the comments, which is $3$, which is not invertible.