Let's solve the problem step by step: --- ### **Step 1: Find the rank of matrix \( A \) and matrix \([A|b]\)** #### **Matrix \( A \):** \[ A = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \] - **Rank of \( A \):** All elements are zero, so the rank is **0**. #### **Augmented Matrix \([A|b]\):** \[ [A|b] = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ \end{bmatrix} \] - **Rank of \([A|b]\):** The first three columns are zero, but the last column has non-zero entries. The rank is **2** (since both rows are linearly independent and non-zero in the last column). **Explanation (Step 1):** - The rank of \( A \) is 0 because all rows are zero. - The rank of \([A|b]\) is 2 because both rows have non-zero entries in the last column, making them linearly independent. --- ### **Step 2: Compare ranks and analyze the solution** - Is \( \text{rank}(A) = \text{rank}([A|b]) \)? - **No** (rank of \( A \) is 0, rank of \([A|b]\) is 2) - Is the solution unique? - **No** (If the ranks are not equal, the system is inconsistent and has no solution.) **Explanation (Step 2):** - Since the ranks are not equal, the system has **no solution** (it is inconsistent). --- ### **Final Answers:** - **rank(A):** 0 - **rank([A|b]):** 2 - **Is rank(A) = rank([A|b])?** No - **Is the solution unique?** No - **Find any solution:** (No solution, so leave the solution fields blank or write "null") --- **Summary:** The system has no solution because the ranks of the coefficient matrix and the augmented matrix are not equal.

Let's analyze and solve the problem step by step: --- ### **Step 1: Find the rank of matrix \(A\) and the augmented matrix \([A|b]\)** #### **Matrix \(A\):** \[ A = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \] - **Explanation:** All entries of matrix \(A\) are zero, which means all rows are zero vectors. Therefore, the rank of \(A\) (the maximum number of linearly independent rows) is **0**. #### **Augmented matrix \([A|b]\):** \[ [A|b] = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ \end{bmatrix} \] - **Explanation:** The first three columns are zeros, and the last column contains \(1\) and \(2\). The two rows are linearly independent because they differ in the last entry, so the rank of \([A|b]\) is **2**. --- ### **Step 2: Determine the nature of the solutions** - **Compare the ranks:** \[ \text{rank}(A) = 0, \quad \text{rank}([A|b]) = 2 \] - **Conclusion:** Since \(\text{rank}(A) \neq \text{rank}([A|b])\), the system is **inconsistent** and **has no solutions**. --- ### **Final Answer:** - **Rank of \(A\):** 0 - **Rank of \([A|b]\):** 2 - **Does the system have a unique solution?** No, because the system is inconsistent. - **Solution status:** No solutions exist. --- **In summary:** The system has **no solution** because the ranks are not equal.