For a quadratic form in $3$ variable over $\mathbb{R}$, let $r$ be its rank and $s$ be its signature. The number of possible pairs $(r,s)$ is
$(a)$ $13$.
$(b)$ $9$.
$(c)$ $10$.
$(d)$ $16$.
Since a quadratic form in $3$ variables is a $3\times 3$ matrix, it has a possible rank of $0,1,2,3$ ($0$ if we allow the quadratic form identically zero, hence null matrix). So The answer should be a multiple of $4$, so option $(d)$ should have been correct. But the answer is given to be $10$. I don't get what's going wrong here. Any help is appreciated.
There are various definitions of "signature" in the literature.One definition given by Wolfram Math World is "the signature is sometimes defined to be the number of positive squared terms in its reduced form" I'll use this definition . Upon diagonalization, rank=number of non-zero terms, signature =number of positive terms. So $$ \text { if }r=3, s=3,2,1, \text { or } 0$$$$ \text { if }r=2, s=2,1, \text { or } 0$$$$ \text { if }r=1, s=1, \text { or } 0$$$$ \text { if }r=0, s= 0$$ Thus there are 10 possibilities for $(r,s)$