Quadratic matrix ( question with parameter)

Question

Let A = $\pmatrix{s&s&0\\ s&s&0\\ 0&0&t}$ , where $s, t \in \mathbb R$ are parameters, and let QA : $R_ 3$ → R be the corresponding quadratic form. Determine for which values of s and t, A and QA are positive semi-definite

Answer 1

The associated quadratic form is $$Q(x,yz)=sx^2+2sxy+sy^2+tz^2=s(x-y)^2+tz^2,$$ hence it is semi-definite positive if and only if ($s\ge 0$ and $t\ge 0$).

Answer 2

Using the method of elemental transformations by rows and columns: $$\left[\begin{array}{ccc|ccc} s & s & 0 \\ s & s & 0 \\ 0 & 0 & t \end{array}\right]\begin{matrix}{R_2-R_1}\end{matrix}\sim \left[\begin{array}{ccc|ccc} s & s & 0 \\ 0 & 0 & 0 \\ 0 & 0 & t \end{array}\right]\begin{matrix}{C_2-C_1}\end{matrix}\sim \left[\begin{array}{ccc|ccc} s & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & t \end{array}\right]$$ So, $A$ is congruent with $D=\text{diag }(s,0,t)$ hence, $A$ is positive semi definite iff $s\ge 0$ and $t\ge 0.$