Question: Let A be a square matrix that can be partitioned as

$$A =\begin{pmatrix}P&0\\Q&S\end{pmatrix}$$ P and S are square matrices. Such a matrix is said to be in block (upper) triangular form. Prove that det = (det P)(det S).

Hint: Try a proof of induction on the number of rows of P

Okay, I was hoping someone could give me a more helpful hint of where to start. I know that induction is where you solve for a base case. Then you assume it is true for n and try to prove that it is true for n +1. If it is true for n +1 then it is true for all numbers from 1 to n. I understand that concept. I just have no idea where to begin.

Answer: 
If $P$ is 1 by 1, then, using the cofactor expansion for the determinant along the first row
$$
\det A = \det \pmatrix{P & 0 \\ Q & S} = \det \pmatrix{p\_{11} & 0 \\ Q & S} = p\_{11}\cdot\det S = \det P \cdot \det S
$$
since the determinant of a 1 by 1 matrix is just the value of its entry. 

Then, assume this holds for matrices $P$ that are $n-1$ by $n-1$. Let us denote $P^{(i,j)}$ to be the matrix $P$ with the $i$th row and $j$th column removed and $Q^{(j)}$ be the matrix $Q$ with the $j$th column removed. Then, if $P$ is $n$ by $n$, we can again expand along the first row
\begin{align\*}
\det A &= \det \pmatrix{P & 0 \\ Q & S} = (-1)^0\cdot p\_{11}\cdot \det \pmatrix{P^{(1,1)} & 0 \\ Q^{(1)} & S} + \cdots +(-1)^n\cdot p\_{1n}\cdot \det \pmatrix{P^{(1,n)} & 0 \\ Q^{(n)} & S} \\
&= (-1)^0 \cdot p\_{11}\cdot (\det P^{(1,1)}\cdot\det S) + \cdots +(-1)^{n-1}\cdot p\_{1n}\cdot (\det P^{(1,n)}\cdot\det S) \\&= \det S\cdot((-1)^0\cdot p\_{11}\cdot \det P^{(1,1)} + \cdots +(-1)^n\cdot p\_{1n}\cdot \det P^{(1,n)})\\&=\det S\cdot \det P
\end{align\*}
Going from the second to the third line, we make use of the fact that $P^{(1,i)}$ are necessarily $n-1$ by $n-1$, so the induction hypothesis applies. Going from the third to the fourth line is just the cofactor expansion of the determinant along the first row. 

Hope this helps! I am relatively new to linear algebra, so if something's off please tell me.
[![enter image description here](https://i.stack.imgur.com/ArrOp.png)](https://i.stack.imgur.com/ArrOp.png)

Another way to prove this question (without using mathematical induction)