[Linear Algebra] prove that A is singular (A is a square matrix)?

[nooonz]

Homework Statement

Let A = BC
where B is 8*3 and C is 3*8
Prove that A is singular.

Homework Equations

A is singular when Det(A)=0

The Attempt at a Solution

When B and C are multiplied the result is A which is an 8*8 matrix. However, top prove it can't be non singular i chose a smaller matrix (2*3) and (3*2) and the result showed that if i chose the same numbers the answer is 0. But how do i prove it without numbers?

 

[CompuChip]

I don't really understand the question, I think.
Yes, if det(A) = 0, then A is singular.
Yes, det(A) = det(B) det(C).

But for arbitrary 8x3 and 3x8 matrices B and C, respectively, it is not true (if det(B) and det(C) are non-zero, so is det(A)).

Have you quoted the complete question?

 

[Ray Vickson]

Homework Statement

Let A = BC
where B is 8*3 and C is 3*8
Prove that A is singular.

Homework Equations

A is singular when Det(A)=0

The Attempt at a Solution

When B and C are multiplied the result is A which is an 8*8 matrix. However, top prove it can't be non singular i chose a smaller matrix (2*3) and (3*2) and the result showed that if i chose the same numbers the answer is 0. But how do i prove it without numbers?

You want to show that the rank of A is less than 8; in other words, you want to show that the number of linearly independent columns of A is less than 8. How is column j of A formed?

RGV

 

[nooonz]

I don't really understand the question, I think.
Yes, if det(A) = 0, then A is singular.
Yes, det(A) = det(B) det(C).

But for arbitrary 8x3 and 3x8 matrices B and C, respectively, it is not true (if det(B) and det(C) are non-zero, so is det(A)).

Have you quoted the complete question?

yes, i quoted the entire question

 

[nooonz]

You want to show that the rank of A is less than 8; in other words, you want to show that the number of linearly independent columns of A is less than 8. How is column j of A formed?

RGV

What do you mean how is column j of A formed? How do i show that the rank is less than 8 without using numbers?

 

[Ray Vickson]

If I told you the answers to these questions, I would be doing the whole question for you. I gave you hints, and will stop there.

RGV

 

[Ray Vickson]

What do you mean how is column j of A formed? How do i show that the rank is less than 8 without using numbers?

OK, I will give a bit more of a hint. You have C = A*B, where C is 8x8, A is 8x3 and B is 3x8. Look at it as a linear transformation on E8 = E^8 (8-dimensional real space). It is composed of two transformations B:E8 --> E3, a linear transformation from an 8-dimensional space to a 3-dimensional one, followed by a transformation A:E3 --> E8, from a 3-dimensional space to an 8-dimensional one. Altogether, C:E8 --> E8 transforms E8 into a subspace C(E8) of E8, and you are asked to show that this transformation is not the identity. What is the dimension of the subspace C(E8)? Remember that we get to C(E8) from a subspace B(E8) of E3.

RGV