Answer if its true or false for each 1If AB C then NullC N

Answer if it\'s true or false for each:

1.If AB = C, then Null(C) ? Null(B).

2.For nxn A, if det(A) ? 0, then the rank of A is n.

3.If A is a 3x3 diagonal matrix, then the standard basis {(1 0 0), (0 1 0), (0 0 1)} is an eigenbasis.

5.For non-singular A, Null(A) = Null(A^-1 )

6.All subspaces of a given vector space have a non-empty intersection

7.For 3x4 A, Col(A) ? R ^4

9.If Av = v for some vector v, then 1 is an eigenvalue of A.

Solution

Post multiple questions to get the remaining answers, since there are lot of subparts

1) The statement is FALSE

2) The statement is TRUE, since if the determinant is non-zero, then all the rows if the matrix are independent of each other, hence the rank of A will be equal to the order of matrix which is n

3) The statement is TRUE

9)

Av = (lambda)v

Hence the eigen value for A will be equal to 1

Therefore the statement is TRUE