SOLUTION: Let A be the matrix of coefficients of a 5 × 7 system of linear equations, A⃗x = ⃗b. Using row operations, you find that A is row equivalent to a matrix in reduced row echelon

Question 1200083: Let A be the matrix of coefficients of a 5 × 7 system of linear equations, A⃗x = ⃗b. Using row operations, you find that A is row equivalent to a matrix in reduced row echelon form with one row of zeroes at the bottom.
(a) What is rank(A)?
(b) How many free variables does the system have?
(c) For the given system how many possible solutions could it have? (Circle all which apply)
1. 0 solutions
2. 1 solution
3. infinite solutions
(d) For the associated homogeneous system A⃗x = ⃗0, how many possible solutions could it have? (Circle all which apply)
1. 0 solutions
2. 1 solution
3. infinite solutions
Answer by GingerAle(43) (Show Source): You can put this solution on YOUR website!
**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

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