Question

Find the inverse of A= {{x,0,0},{1,y,0},{0,1,z}} by first factoring out the diagonal entries.

How do I "factor out the diagonal entries?" before finding the inverse?

See picture here (question #3)
http://imgur.com/a/lP0QW

Answer 1

Seems to be a specific type of diagonalization of \(\mathbf A\) by means of row operations. (In case you're not familiar, diagonalization of a matrix is a way to "factorize" a matrix and decompose it into simpler parts, or at the least into parts that make other computations easier.)

You're asked to write \(\mathbf A\) as a product of two "factors" \(\mathbf B\) and \(\mathbf C\), where
\[\mathbf {BC}=\begin{pmatrix}x&0&0\\0&y&0\\0&0&z\end{pmatrix}\begin{pmatrix}1&0&0\\\square&1&0\\0&\square&1\end{pmatrix}\]You can find the missing entries by multiplying to get
\[\mathbf{BC}=\begin{pmatrix}x&0&0\\\square y&y&0\\0&\square z&z\end{pmatrix}=\begin{pmatrix}x&0&0\\1&y&0\\0&1&z\end{pmatrix}=\mathbf A\]