The Last Pivot and Its Multiplier

A numerical solver has to solve $A\mathbf{x} = \mathbf{b}$ for hundreds of different right-hand sides $\mathbf{b}$, so instead of eliminating from scratch every time it factors $A$ once and stores the two triangular pieces.

$A = \begin{bmatrix} 3 & 2 & -1 \\ 6 & 1 & 4 \\ -9 & 3 & 5 \end{bmatrix}$

The factorization is $A = LU$, computed by ordinary elimination with no row swaps, using the standard convention:

So $\ell_{32}$ is the multiplier that clears the entry in row 3, column 2, and $u_{33}$ is the third pivot — the bottom-right entry of $U$.

Report the value of $u_{33} + \ell_{32}$. Your answer is an exact integer.