Question

Find the rank of a 5 x 6 matrix A for which $Ax=0$ has a two-dimensional solution space.

• A. 4
• B. 5
• C. 8
• D. 6

Answer 1

Answer: (A)

4

Answer 2

The problem asks us to find the rank of a 5x6 matrix A, given that the homogeneous system $Ax=0$ has a two-dimensional solution space.

Let A be an $m \times n$ matrix. In this case, $m=5$ (number of rows) and $n=6$ (number of columns).

The solution space of $Ax=0$ is also known as the null space of A, denoted as $N(A)$. The dimension of the null space is called the nullity of A, i.e., nullity(A) = dim($N(A)$). According to the problem statement, the solution space is two-dimensional, so nullity(A) = 2.

The Rank-Nullity Theorem states that for any matrix A, the sum of its rank and nullity is equal to the number of columns of the matrix. Mathematically, this is expressed as:

rank(A) + nullity(A) = n

where n is the number of columns of A.

Substitute the given values into the Rank-Nullity Theorem:

Here, n = 6 and nullity(A) = 2.

rank(A) + 2 = 6

Solving for rank(A):

rank(A) = 6 - 2

rank(A) = 4

The rank of a matrix A is the maximum number of linearly independent rows or columns. For an $m \times n$ matrix, the rank can be at most $\min(m, n)$. In this case, $\min(5, 6) = 5$. Our calculated rank of 4 is consistent with this property, as $4 \le 5$.