It is a linear algebra problem, I have answers but I need a step-by-step solutio

Question

It is a linear algebra problem, I have answers but I need a step-by-step solutions with explonation how to deal with problems like that.

answers: 1) NO 2)0 3)NO 4)YES 5)nullity=2 6)NO 7)YES 8)NO

5. Suppose that A is a 6 x 6 matrix of rank 4. l) Is A non-singular? 2) What can be said about the determinant of A? 3) Do the rows of A span R6? 4) Are there any nontrivial solutions to homogeneous system Ax 0? 5) What is the nullity of A? 6) Is A row equivalent to 16? 7) Does AT exist 8) Can the columns of A be linearly independent vectors in R6?

Explanation / Answer

1) No

note that rank of a singular matrix of order nxn is strictly less than n.So A is singular.

2) 0

A is singular and rank of a A =4,which is strictly less than 6

So determinant != 0

3) no

if A is singular, the echelon form of A is withatleast a zero row, implies the rows of A are linearly dependent.

4) yes

the homogeneous system AX = 0 is with onlytrivial solution if A is nonsingular.

given A is singular , so the homogeneous systempossesses nontrivial solution.

5) Dimension Theorem formula 4: if A is a matrix with n columns, then
rank(A) + nullity(A) = n

So 4 +  nullity(A) =6

nullity(A) = 2

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