We have learned in our earlier article on Block Designs, about the C-matrix and the D-matrix. Today, I will discuss some properties of these matrices.

**Result 1: ** **C is determined entirely by the incidence matrix N.**

**Result 2: C is symmetric, non-negative definite, and positive semi-definite with at least one eigen value equal to 0.**

**Result 3: The reduced normal equations are consistent irrespective of the rank of C.**

**Proof**: We can easily see that:

Now, , which implies Q belongs in the column space of C, completing the proof.

**Result 4: , under .**

**Result 5: under .**

**Result 6: Rank(X’X)=b+Rank(C).**

**Proof:**

Let

Since rank remains unchanged by pre- or post- multiplication by a non singular matrix, consider UAU’ which will have same rank as A.

, Thus Proved.

**Properties of D-matrix:**

- D is symmetric and .

So we can see: b+rank(C)=v+rank(D).

Thus for a block design error degrees of freedom is (n-rank(X)) or (n-b+rank(C)).