True or False If A is an invertible n times n matrix then co

True or False: If A is an invertible n times n matrix, then colspace(A) = rowspace(A). True or False: If A is an n times n matrix, such that the nullity of A is 0, then colspace(A)= R^n. True or False: If A is a 5 times 3 matrix and rank(A) = 3, then row space(A) = R^n. True or False: If A is a 3 times 3 matrix and rank(A) = 3, then there exists a vector b in R^3 such that the augmented matrix [A|b] has rank 2. True or False: If A is a 3 times 3 matrix and rank(A) = 2, then there exists a vector b in R^3 such that the augmented matrix [A|b] has rank 3. True or False: If A is a 5 times 3 matrix and rank(A) = 3, then there exists a vector b in R^5 such that the augmented matrix [A|b] has rank 4. True or False: There exists a 5 times 3 matrix A with rank(A) = 3 and nullity(A) = 2. True or False: There exists a 3 times 5 matrix A with rank(A) = 3 and nullity(A) = 2. True or False: All (finite dimensional) vector spaces must have exactly one basis. True or False: If A is a 4 times 4 matrix then the Rank(A) = Nullity(A).

Solution

1) True If A is an invertible matrix of order nxn, then Colspace (A)=Row space(A)

2) True

3) False

4) False

5) False if b belongs to Colspace(A) and True if b belongs to Rowspace(A)

6) False

7) False rank of A=nullity(A)

8) False

9) False

10) False nullity(A)=0 then Rank(A)=n