Finding the LU decomposition of a matrix using the shortcut method

Question

I was asked to find the LU decomposition of

$$\begin{bmatrix}5&4\\-2&-3\\\end{bmatrix}$$

I know that the shortcut method means finding the upper and using the multiplier to find the lower. In this case, you cannot multiply by one number, but you need to multiply both row one and row two. Does this mean I cannot use this method? What would be the suggested method for finding the LU decomposition?

Answer 1

Hint:

By Gaussian elimination,

$$\begin{bmatrix}5&4\\-2&-3\end{bmatrix}\stackrel{\times\frac15}\to\begin{bmatrix}1&\frac45\\-2&-3\end{bmatrix}\to\begin{bmatrix}1&\frac45\\0&-\frac75\end{bmatrix}\stackrel{\times-\frac75}\to\begin{bmatrix}1&\frac45\\0&1\end{bmatrix}.$$

This contains all the coefficients of the LU decompostion.