2. Consider the following statements about the rank of a matrix. (I) If matrix B

Question

2. Consider the following statements about the rank of a matrix. (I) If matrix B can be obtained from matrix A by a sequence of elementary row operations, then rank(A) = rank(B) (II) The rank of a matrix A is the number of non- zero rows in A (II) If the mxn matrix A is the coefficient matrix of a system of linear equations and the number of free variables in that systen is denoted by f, then rank(A)+ f Which of statements (T), (II) and (III) are neces- sarily true? (a) Only (I) (e) Only (1) and (II) (d) Only (1) and (III) (e) (I), (II) and (III) (b) Only (I)

Explanation / Answer

Since elementary operations does not change the rank of the matrix so, Rank(A)=Rank(B) is true in always. If B is obtained from A after finite number of elementary operations.

So Statement I is always true

(II) Rank of the matrix is the number of nonzero rows in the row-reduced Echlon form of A, not of A

So statement II is not true in general

(III) Since Rank(A) is less or equal to minimum of m and n,

And If there are f free varaibles then row reduced echlon form of A has n-f nonzero rows,

That is rank is n-f, That is Rank(A)+f=n

Hence Statement III is correct always

Therefore Optoin D is correct

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