[Sticky] Properties of Determinants
With an FP2 test on the first part of Matrices, given that I've forgotten a lot about this, I need to revise these. Having said that a good post on the matter will solidify these in my head.
- Swapping two columns of a matrix reverses the sign of the determinant. Picturing the little matrix of cofactors helps me visualize this personally. Since cofactor matrix "prqs" has det = ps-qr, and swapping the rows gives you matrix "rpsq" with det = rq-sp (which is the first determinant multiplyed by -1).
- If two columns of a Matrix are identical, it's determinant is equal to 0. Pretty straight forwards if you think of this in terms of the first property. If you swap the identical columns the determinant remains the same (but it's supposed to be multiplied by -1) therefore it must be 0.
- Cyclic interchange of columns in a Matrix leaves it's determinant unchanged. This is because it involves an even amount of row swaps- meaning that the sign is changed back to the original always.
- Expanding by an alien cofactor always gives the answer 0. a1B1 + a2B2 + a3B3 = 0
- The determinant of a 3x3 Matrix is the volume scale factor of the transformation it defines. Just like the det of a 2x2 matrix is the area scaled factor of transformation.
- The determinant of a product of 2 square matrices is the product of the two separate matrices. det (MN) = det (M) x det (N)
- Cofactors Alternate sign.
PlanckTime - Amin
Don't forget Sarrus' method to find the determinant.
Det (M) =RHS-LHS
RHS is the sum of the diagonals from the left side, down
And LHS is the diagonals coming in and down from the right hand side.
When a diagonal goes beyond the bottom, of the matrix, it reappears at the top.
Its a much quicker way to do the determinant, but it is also less useful because it doesnt provide as much information
"Either this wallpaper goes, or I go"- Oscar Wilde's last words
Thanks mate, I guess it's not really a property of the determinants, but I agree best method for finding it!
PlanckTime - Amin