How are the determinants of A and B related? (Do not compute det(A))

[Incognitopad]

Homework Statement

Consider the matrices

A =
a1 a2 a3
b1 b2 b3
c1 c2 c3
and

B =
3a1 4a2+5a1 6a3
3b1 4b2+5b1 6b3
3c1 4c2+5c1 6c3

How are the determinants of A and B related? DO NOT COMPUTE det(A)!

Homework Equations

The Attempt at a Solution

I'm completely lost... I tried doing Kramer rule for det(B) but don't understand how I'm supposed to do this. I mean, I can see that the 2nd column of B is equal to 5 times the first plus 4 times the second columns of A, that the 1st column of B is 3 times the first column of A and that the 3rd column of B is 6 times the third column of A. But what am I meant to draw from this?

 

[jedishrfu]

I defer to D H

 

[D H]

I mean, I can see that the 2nd column of B is equal to 5 times the first plus 4 times the second columns of A, that the 1st column of B is 3 times the first column of A and that the 3rd column of B is 6 times the third column of A. But what am I meant to draw from this?

Almost there. Try to express this in matrix form. In other words, see if you can find a matrix C such that either C*A=B or A*C=B.

What's the determinant of a product of two matrices?

 

[Ray Vickson]

Homework Statement

Consider the matrices

A =
a1 a2 a3
b1 b2 b3
c1 c2 c3
and

B =
3a1 4a2+5a1 6a3
3b1 4b2+5b1 6b3
3c1 4c2+5c1 6c3

How are the determinants of A and B related? DO NOT COMPUTE det(A)!

Homework Equations

The Attempt at a Solution

I'm completely lost... I tried doing Kramer rule for det(B) but don't understand how I'm supposed to do this. I mean, I can see that the 2nd column of B is equal to 5 times the first plus 4 times the second columns of A, that the 1st column of B is 3 times the first column of A and that the 3rd column of B is 6 times the third column of A. But what am I meant to draw from this?

You are supposed to use standard results about determinants obtained by adding multiples of row (or columns) to other rows (or columns). I will let you find these, because the exercise of searching will help you to remember them.

 

[Incognitopad]

oh, is it that simple?

3*5*6det(A)=det(B)?

 

[Ray Vickson]

oh, is it that simple?

3*5*6det(A)=det(B)?

Almost. Where does the factor '5' come from?

 

[Incognitopad]

sorry. typo. i meant to write a 4. adding columns/rows into others don't change the determinant, but multiplying a row/column multiplies the determinant by the same factor.

3*4*6

72det(A)=det(B)

 

[Ray Vickson]

sorry. typo. i meant to write a 4. adding columns/rows into others don't change the determinant, but multiplying a row/column multiplies the determinant by the same factor.

3*4*6

72det(A)=det(B)

Right!