Question 1200083
**a) Rank(A)**

* The rank of a matrix is the number of non-zero rows in its row-echelon form. 
* Since A has one row of zeros and is row equivalent to a matrix in reduced row echelon form, its rank is 4.

**b) Number of Free Variables**

* The number of free variables is equal to the number of columns minus the rank of the matrix.
* Number of free variables = 7 (columns) - 4 (rank) = 3

**c) Possible Solutions for A⃗x = ⃗b**

* **Possible Solutions:**
    * **Infinite solutions** 
    * **No solutions**

* **Explanation:**
    * If the last row in the reduced row echelon form of the augmented matrix [A | ⃗b] is of the form [0 0 0 | c] where c is a non-zero constant, then the system has no solution. 
    * Otherwise, if the last row is all zeros, the system will have infinite solutions due to the free variables.

**d) Possible Solutions for A⃗x = ⃗0**

* **Possible Solutions:**
    * **Infinite solutions**

* **Explanation:**
    * For the homogeneous system A⃗x = ⃗0, the last row in the augmented matrix [A | ⃗0] will always be all zeros. 
    * Since there are free variables, the homogeneous system will always have infinite solutions (including the trivial solution ⃗x = ⃗0).

**In Summary:**

* **Rank(A) = 4**
* **Number of Free Variables = 3**
* **Possible Solutions for A⃗x = ⃗b:** Infinite solutions or no solutions
* **Possible Solutions for A⃗x = ⃗0:** Infinite solutions