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 solution space of the homogeneous equation $Ax = 0$ is two-dimensional.

Let A be an $m \times n$ matrix. In this case, A is a 5x6 matrix, so $m=5$ and $n=6$.

The solution space of $Ax = 0$ is also known as the null space (or kernel) of the matrix A, denoted as N(A). The dimension of the null space is called the nullity of A, denoted as nullity(A).

Given: The solution space of $Ax = 0$ has a two-dimensional solution space. This means nullity(A) = 2.

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

Rank(A) + Nullity(A) = n

In this problem: Number of columns, n = 6 Nullity(A) = 2 (given)

Substitute these values into the Rank-Nullity Theorem: Rank(A) + 2 = 6

Now, solve for Rank(A): Rank(A) = 6 - 2 Rank(A) = 4

Thus, the rank of the matrix A is 4.