Q:

Why am I not getting two positive pivots for this symmetric matrix?

I'm watching Gilbert Strang's linear algebra lecture on positive definite matrices and I have a question about the example he uses in the last 10-15 minutes of the video.
Strang is demonstrating the claim that all pivots of a positive definite matrix are positive using the example of
$$\mathbf{A}=\begin{bmatrix}5&2\\2&3\end{bmatrix}$$
He finds that the pivots are $5$ and $\frac{11}{5}$, but I'm not sure how he gets the second one. Elimination $(2\text{row}_1-5\text{row}_2\to\text{row}_2)$ yields for me
$$\begin{bmatrix}5&2\\0&-11\end{bmatrix}$$
which suggests to me that the pivots are $5$ and $\color{red}{-11}$. But that can't be right, as Strang goes on to say that the product of the pivots is equal to the determinant of $\mathbf{A}$, and indeed: if one of the pivots is $5$ (no argument there), this means $5x=11\implies x=\frac{11}{5}$. Furthermore, the eigenvalues of $\mathbf{A}$ are both positive $(4\pm\sqrt5)$ which means both pivots must be positive.
How do I get $\frac{11}{5}$? Am I misunderstanding something fundamental about what a pivot is?

A:

Strang doesn't multiply rows until the end. His row operation is subtracting $2/5$ of row $1$ from row $2$ to get
$$
\pmatrix{5&2\\ 0&11/5}
$$
Note that such row operations do not affect the determinant. Note also that the other row operations, i.e. row-switching and multiplying rows by scalars do affect the determinant.
Your row operation is tantamount to doing this and then multiplying the second row by $-5$.
It seems that for GS, pivots in this context refer to the entries on the diagonal after row reducing using only the operation of adding one multiple of a row to another.

