# Proof of det of a matrix with submatrices

1. Apr 11, 2012

### Hernaner28

1. The problem statement, all variables and given/known data
PROBLEM 1:
Let A,B,C,D four matrices nxn which are submatrices of matrix 2nx2n:
$$E = \left( {\begin{array}{*{20}{c}} A&B\\ C&D \end{array}} \right)$$

PROBLEM 2:

Let A a matrix nxn, B a matrix nxm and C a matrix mxm. O is the null matrix mxn.

2. Relevant equations
PROOF 1:
Say whether it's false or true that:
$$\det (E) = \det (A)\det (D) - \det (B)\det (C)$$

PROOF 2:
Proof that:
$$\det \left( {\begin{array}{*{20}{c}} A&B\\ O&C \end{array}} \right) = \det (A)\det (C)$$

3. The attempt at a solution
No idea how to start. I know this is true for numbers but how can I make it generic for matrix? Just an idea?

Thanks!!

2. Apr 11, 2012

### Dick

3. Apr 11, 2012

### Hernaner28

Well, we defined it first by taking the column numbers:

$$|A| = {( - 1)^{1 + 1}}{a_{11}}\left| {{A_{11}}} \right| + ... + {( - 1)^{i + 1}}{a_{i1}}\left| {{A_{i1}}} \right| + ... + {( - 1)^{n + 1}}{a_{n1}}\left| {{A_{n1}}} \right|$$

also for rows..

Thanks!

BTW, what is this for? This is not a definition for submatrices but numbers.

Last edited: Apr 11, 2012
4. Apr 12, 2012

### Hernaner28

No idea?

5. Apr 12, 2012

### HallsofIvy

That's not a definition until you tell what $|A_{ij}|$ means! And, you cannot define them as determinants in a definition of "determinants".

Do you believe it is not necessary to know what a "determinant" is to prove something about a determinants?

6. Apr 12, 2012

### Hernaner28

So, what's the definition? That's the one I have on my book. Just copied it.

Thanks!

PD: We defined the determinant of a matrix 2x2 as ad-bc and then we defined a 3x3 matrix and so on...

7. Apr 12, 2012

### Dick

I was hoping you had a definition more like http://en.wikipedia.org/wiki/Determinant in the "nxn matrices" section. You sum the products of elements from every row and column times a permutation factor. If you apply that to proof 2 you can see that none of the nonzero contribution to the determinant comes from B. So you may as well put B=0 as well. Now you can reduce the matrix to upper triangular without mixing A and C.

It should be easy enough to find a counterexample for Proof 1.