# Thread: 3 x 3 Matrix

1. ## 3 x 3 Matrix

Consider the following formula for the determinant of a 3 x 3 matrix:

det(x1y1z1//x2y2z2//x3y3z3) = x1y2z3+y1z2x3+z1x2y3 - (x3y2z1+y3z2x1+z3x2y1)

a) Show that is it linear in the first column.
b) Show that it is alternating when you switch the first and second column.

(// denotes the next row in the matrix)

I'm not sure how to prove this, any help is great. Thanks.

2. Originally Posted by Fel
Consider the following formula for the determinant of a 3 x 3 matrix:

det(x1y1z1//x2y2z2//x3y3z3) = x1y2z3+y1z2x3+z1x2y3 - (x3y2z1+y3z2x1+z3x2y1)

a) Show that is it linear in the first column.
b) Show that it is alternating when you switch the first and second column.

(// denotes the next row in the matrix)

I'm not sure how to prove this, any help is great. Thanks.
a) It says that if you multiply the first column by a number k, then the determinant also get multplied by k. To prove this, multiply x1,x2,x3 by k and see what happens to determinant.

3. Originally Posted by malaygoel
a) It says that if you multiply the first column by a number k, then the determinant also get multplied by k. To prove this, multiply x1,x2,x3 by k and see what happens to determinant.
It also says that if you replace x1, x2, x3 by a1+ b1, a2+ b2, a3+ b3, you will get the determinant with a1, a2, a3 plus the determinant with b1, b2, b3.

Tedious calculation but just basic algebra.