Proof of determinant properties

[eyehategod]

I need help with proof. Can anyone lead me in the right direction?
http://i153.photobucket.com/albums/s208/nailbomb2/LastScan.jpg

 

[Dick]

The determinant is a multilinear function of the columns. That means every term in the polynomial expansion of the determinant contains exactly one entry from the first column. How's that for a direction? Think about expanding by minors.

 

[eyehategod]

Im still lost. I am not understanding.

 

[robphy]

A related problem (actually a special case)... concerning cross-products involving vectors A, B, and M.(A x M) + (B x M) = (A+B) x M

Can you do the 2x2 version of your question? The 1x1 is easy.

 

[HallsofIvy]

The determinant is a multilinear function of the columns. That means every term in the polynomial expansion of the determinant contains exactly one entry from the first column. How's that for a direction? Think about expanding by minors.

Im still lost. I am not understanding.

Do you know how to "expand by minors"? If so find the determinant on the right by expanding along the first column.

 

[Dick]

Look up 'expansion by minors'. You can turn the first 3x3 determinant into a sum of three 2x2 determinants a_11*A_11-a_21*A_21+a_31*A_31, where A_ij is a 2x2 determinant that doesn't include any elements of the first column. The second one is b_11*A_11-b_21*A_21+b_31*A_31. The important thing is that the A determinants are the same.