A is a 3 × 5 real matrix of rank 2. For the set of homogeneous equations Ax = 0, where 0 is a zero vector and x is a vector of unknown variables, which of the following is/are true?

Correct Options:
1. The given set of equations will be satisfied by a zero vector of appropriate size.
2. The given set of equations will have infinitely many solutions.

Explanation:

We are given a homogeneous system of linear equations represented by:
$$Ax=0$$
where $A$ is a real matrix of size $3\times 5$ and its rank is $\text{rank}\left(A\right)=2$.

Step 1: Analyzing the size of the vectors
Since $A$ has dimensions $3\times 5$, it has $3$ rows and $5$ columns. For the matrix multiplication $Ax$ to be defined, the vector of unknowns $x$ must be a column vector of size $5\times 1$.
A homogeneous system of the form $Ax=0$ is always consistent and is always satisfied by the trivial solution, which is the zero vector:
$$x={[0,0,0,0,0]}^T$$
Therefore, the statement "The given set of equations will be satisfied by a zero vector of appropriate size" is correct.

Step 2: Determining the number of solutions using the Rank-Nullity Theorem
According to the Rank-Nullity Theorem, the sum of the rank and the nullity of a matrix equals the number of columns (variables) $n$ of that matrix:
$$\text{Rank}\left(A\right)+\text{Nullity}\left(A\right)=n$$
Here, the number of columns (unknown variables) is $n=5$, and the rank is $\text{Rank}\left(A\right)=2$. Substituting these values into the theorem:
$$2+\text{Nullity}\left(A\right)=5$$
$$\text{Nullity}\left(A\right)=5-2=3$$
The nullity represents the dimension of the null space, which is the number of free variables in the system. Since the nullity is $3$ (which is greater than $0$), there are $3$ free variables in the solution.

Step 3: Conclusion
Because there are free variables and we are dealing with a real vector space, we can assign any real value to these free variables. This means the system has non-trivial solutions, and because the field of real numbers is infinite, the system of equations must have infinitely many solutions.
Thus, the statement "The given set of equations will have infinitely many solutions" is also correct.