Proof of det of a matrix with submatrices

[Hernaner28]

Homework Statement

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.

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

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!

 

[Dick]

What's your definition of det?

 

[Hernaner28]

What's your definition of det?

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.

 

[Hernaner28]

No idea?

 

[HallsofIvy]

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

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

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

also for rows..

Thanks!

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

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

 

[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...

 

[Dick]

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...

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.