Find the missing variance in a $2×2$ covariance matrix

Question

Given the covariance matrix:

$\sum = \begin{pmatrix} 1 & 2 \\ 2 & b \end{pmatrix}$

I can't seem to figure out what the value or range of $b$ is. Intuition is telling me $4$ but I am not sure. Thanks in advance.

Answer 1

We know that the covariance matrix must always be positive semi-definite, which means that for any vector $x=\begin{bmatrix} u \\ v \end{bmatrix}$ with $u,v\in \Bbb{R}$, we must have $$\begin{aligned} x^T\Sigma x\ge 0 & \iff \begin{bmatrix} u& v \end{bmatrix}\begin{bmatrix} 1& 2 \\ 2& b \end{bmatrix}\begin{bmatrix} u \\ v \end{bmatrix} \ge 0 \\ &\iff u^2+4uv+bv^2 \ge 0 \\ & \iff (u+2v)^2+(b-4)v^2\ge 0 \end{aligned}$$

Taking $u=-2v$, which we can take since the inequality must hold for all $u,v\in \Bbb{R}$, we get that $$(b-4)v^2\ge 0 \iff (b-4)\ge 0 \iff b\ge 4$$ so you do get a range on $b$.