A is row equivalent to B 1 7 2 2 2 6 0 0 11 3 1 3 0 0 0 1 6

A is row equivalent to B = [1 -7 2 2 2 6 0 0 11 3 1 -3 0 0 0 1 6 0], find rank A & dim Nul A For a 4 Times 7 matrix A if rank A is 2, find dim Nul A. If A is n Times n with rank A = n show that Nul A = {0}. Explain why A^-1 exists.

Solution

(a) Since matrices A and B are row equivalent, so both have same properties.

rank A = rank B = number of non-zero rows = 3

Since, we know that

rank B + nullity B = number of columns of B

3 + nullity B = 6

nullity B = 6 - 3 = 3

Therefore, dimension (Nul A ) = nullity B = 3

(b) Since A is 4x7 matrix and its rank is 2.

Since, we know that

rank A + nullity A = number of columns of A

2 + nullity A = 7

nullity A = 7 - 2 = 5

Therefore, dimension (Nul A ) = nullity A = 5

(c) Since A is nxn matrix whose rank is n.

So, A is nonsingular.

using formula

rank A + nullity A = number of columns of A

n + nullity A = n

nullity A = n - n ={ 0}

Therefore, dimension (Nul A ) = nullity A = 0

Since A is nonsingular, therefore inverse of A exists.