For which of the following matrices number of left inverses is greater than the number of right inverses?

Question

Let A be an m × n matrix. If there exists a matrix L of type n × m such that LA = I_n, then L is called left inverse of A. Similarly, if there exists a matrix R of type n × m such that AR = I_m, then R is called right inverse of A.

For example, to find right inverse of matrix

⇒ x - u = 1

y - v = 0

z - w = 0

x + u = 0

y + v = 1

z + w = 0

2x + 3u = 0

2y + 3v = 0

2z + 3w = 1

As this system of equations is inconsistent, we say there is no right inverse for matrix A

For which of the following matrices number of left inverses is greater than the number of right inverses?

Answer 1

Answer is (C)

By observation, there cannot be any left inverse for (B) and (D), so we will check for (A) and (C) only

For (A) let left inverse be

Now, a - 3b = 1, 2a + 2b = 0 and 4a + b = 0, which is not possible. For (C)

Therefore, there are infinite number of left inverses Right inverse:

⇒ a + 4d = 1, 2a - 3d = 0 and 5a + 4d = 0

which is not possible.

Therefore, there is no right inverse