Finding the Determinant of a 4x4 Matrix with Variable Rows

[ahsanxr]

Homework Statement

The 4x4 matrix with rows v1, v2, v3 and v4 has a determinant of -5. What is the determinant of the matrix with rows v1, v2, 7v3+6v4, 6v3+8v4?

Homework Equations

The Attempt at a Solution

I tried doing -5x7x8=-280 but its saying its wrong. I don't understand why. I'm using the properties of determinants of matrices.

 

[Dick]

You are using the linearity of determinants, right? What about the contribution from [v1, v2, 6v4, 6v3]?

 

[ahsanxr]

Yes but the property I read was that if a multiple of a row is added to another row, then there is no change to the determinant.

 

[Dick]

Yes but the property I read was that if a multiple of a row is added to another row, then there is no change to the determinant.

True. But you added a multiple of v4 to the third row. That didn't change the determinant, but it did change the third row. Now when you add a multiple of v3 to the fourth row, you can't claim that doesn't change it. Because the third row isn't v3 anymore! Use linearity directly.

 

[ahsanxr]

v3 still remains v3 because it is not defined as the 3rd row of a matrix in general but instead is the specific 3rd row of the original matrix.

 

[Dick]

v3 still remains v3 because it is not defined as the 3rd row of a matrix in general but instead is the specific 3rd row of the original matrix.

That's wrong. I'm not going to argue with you why. Use linearity. Stuff like det[v1,v2,a*v3+b*v4,v4]=det[v1,v2,a*v3,v4]+det[v1,v2,b*v4,v4], and you'll see why your answer is wrong. Apply it to det[v1,v2,a*v3+b*v4,c*v3+d*v4]. There are TWO nonvanishing determinants in the expansion.

 

[ahsanxr]

I don't understand by what you mean by "linearity." I haven't heard of that term. I was taught the properties of determinants but that's it.

 

[Dick]

I don't understand by what you mean by "linearity." I haven't heard of that term. I was taught the properties of determinants but that's it.

Huh. I would have listed linearity first in my list of determinant properties. It's what I tried to sketch in the last post. If a row of a matrix is given by the sum of two vectors A+B, then the resulting determinant is the sum of the determinant of the matrix with the row replaced by A and the determinant of the matrix with the row replaced by B.

 

[ahsanxr]

Oh that was the name of a property. I didn't know that. I got the right answer now. Thanks for your help.