Question

State whether each of the following sets of data are possible for the matrix equation AX = B. If possible, describe the solution set. That is, tell whether there exists a unique solution, no solution or infinitely many solutions. Here, [A|B] denotes the augmented matrix.

(A) A is a 5 × 6 matrix, rank(A) = 4 and rank[AIB] = 4.
(B) A is a 3 x 4 matrix, rank (A) = 3 and rank [AIB] = 2.
(C) A is a 4 × 2 matrix, rank (A) = 4 and rank[A|B| = 4
(D) A is a 5 × 5 matrix, rank (A) = 4 and rank [A|B] = 5.
(E) A is a 4 × 2 matrix, rank (A) = 2 and rank [A|B] = 2.

Answer 1

Answer:

(A) Possible, the system will always have infinite solutions.

(B) Impossible.

(C) Impossible.

(D) Possible, there is no solutions.

(E) Possible, unique solution.

Step-by-step explanation:

(A) It possible, for example consider

$\left[\begin{array}{cccccc|c}1&0&0& 0 & 0& 0& 0\\0&1&0&0 & 0 & 0 & 0\\0&0&1&0 & 0 & 0& 0\\0&0&0&1 & 0 & 0& 0\\0&0&0&0& 0 & 0& 0\end{array}\right]$

in this case, the system will always have infinite solutions (there at most two variable that could take any valour).

(B) It is impossible because $rank[A|B] \ge 3$.

(C) It is impossible because $rank(A) \le 2$.

(D) It is possible, for example consider

$\left[\begin{array}{ccccc|c}1&0&0& 0 & 0& 0\\0&1&0&0 & 0 & 0 \\0&0&1&0 & 0 & 0\\0&0&0&1 & 0 & 0\\0&0&0&0& 0 & 1\end{array}\right]$

but in this case there is no solutions.

(E) It is possible, for instance, consider

$\left[\begin{array}{cc|c}1&0&0\\0&1&0\\0&0&0\\0&0&0\end{array}\right]$

Now, if this is the case there exists a unique solution.