How to find the inverse of a non-singular block matrix?

Question

Let $P$ a 3x4 matrix. Augmenting $P$ by one row to make a 4x4 non-singular matrix, denoted $P^{*}$. Now letting $H = P^{*-1}$. What is the result of $PH$?

Answer 1

N.B. When you post a question, you should provide some context or details about what you have already tried so that it is easier for others to help; I will answer your problem assuming you have done nothing yet.

It may be helpful to break down the product of the two matrices in terms of each row of matrix $P$ being right-multiplied by the matrix $H = P^{*-1}.$ Consider for instance the first row of $P.$ Obviously it is equal to the first row of $P^*.$ We also know that the product of the first row of $P^*$ with $P^{*-1}$ (what is that matrix product?) is $\begin{pmatrix} 1 & 0 & 0 & 0 \end{pmatrix}.$ You can extend this idea to the other rows. In the end you will get exactly what you expect:

$$\boxed{\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}}.$$

Hope this helps!