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Consider the following assertions: I. Rank (ST) = Rank S = Rank T II. Rank (ST) = Rank S, if T is non-singular III. Rank (ST) = Rank T, if T is non-singular Where S, T : V → V are linear transformations of a finite dimensional vector space V. Which of these is/are correct?

Consider the following assertions: I. Rank (ST) = Rank S = Rank T II. Rank (ST) = Rank S, if T is non-singular III. Rank (ST) = Rank T, if T is non-singular Where S, T : V → V are linear transformations of a finite dimensional vector space V. Which of these is/are correct? Correct Answer Only II

Explanation:

From the properties of a rank of a matrix,

Let A be an m × n matrix, if P and Q are invertible m × m and n × n matrices respectively. Then

Rank (AQ) = Rank (A) = Rank (PA)

Now check the given statements,

I. Rank (ST) = Rank S = Rank T; there is no information of inverse of matrices so it is not necessarily true

II. Rank (ST) = Rank S, if T is non-singular; given that T is non-singular (i.e. invertible) hence it is true

III. Rank (ST) = Rank T, if T is non-singular; it is true when S is invertible but there is no information about the inverse of the matrix.

Hence only second statement is true.

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