# Homework Help: Determinant proof

1. Dec 10, 2007

### eyehategod

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2. Dec 10, 2007

### 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.

3. Dec 11, 2007

### eyehategod

Im still lost. Im not understanding.

4. Dec 11, 2007

### 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.

5. Dec 11, 2007

### HallsofIvy

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

6. Dec 11, 2007

### 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.