GO Classes DPP | GATE CS, DA | Linear Algebra | System of Linear Equations

Answer 1

$$
\begin{aligned}
&Ax=b,\quad A\in\mathbb{R}^{m\times n},\quad b\in\mathbb{R}^{m},\quad m>n.\\
&\text{Since }m>n,\text{ the system is overdetermined.}\\
&\text{If }\operatorname{rank}(A)=\operatorname{rank}([A|b])=n,\text{ there is a unique solution.}\\
&\text{If }\operatorname{rank}(A)=\operatorname{rank}([A|b])<n,\text{ there are infinitely many solutions.}\\
&\text{If }\operatorname{rank}(A)<\operatorname{rank}([A|b]),\text{ the system is inconsistent, hence no solution.}
\end{aligned}
$$

$$
\therefore\ \boxed{A,\ B,\ D}
$$

Answer 2

A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. 

A system of linear equations with more equations than unknowns can be consistent with unique or infinite solutions as well as it could be inconsistent because to make number of equations more than the number of unknowns we can always add any equation which is  linear combination of existing set of equations.  

---

A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system.  

An underdetermined system always has more variables than equations. There cannot be more basic variables than there are equations, so there must be at least one free variable. Such a variable may be assigned infinitely many different values. If the system is consistent, each different value of a free variable will produce a different solution. So, An underdetermined system can NEVER be "Consistent with Unique Solution".

Answer 3

1️⃣ Overdetermined System

An overdetermined system means:

• Number of equations > number of variables

Example:

{x+y=22x+y=33x+y=5\begin{cases} x + y = 2 \\ 2x + y = 3 \\ 3x + y = 5 \end{cases}⎩⎨⎧​x+y=22x+y=33x+y=5​

Here 3 equations, 2 variables → overdetermined.

---

2️⃣ Non-homogeneous System

A non-homogeneous system means the right side is not zero.

Example:

Ax=bAx = bAx=b

where b ≠ 0.

---

3️⃣ What happens in an overdetermined system?

Since there are more equations than variables, all equations usually cannot be satisfied simultaneously.

So generally the system becomes inconsistent.

Example:

x=1x = 1x=1 x=2x = 2x=2

Both cannot be true at the same time.

Thus no solution exists.

---

4️⃣ Correct Option

D. inconsistent having no solution

---

✔ Final Answer: D

---

🧠 Quick exam intuition

System type	Meaning	Solution
Underdetermined	equations < variables	many solutions
Square system	equations = variables	unique solution
Overdetermined	equations > variables	usually no solution