3x3 determinants + factorization

Notes on 3x3 determinant

• - If M is a 3x3 matrix , the determinant of M is the volume scale factor (note 2x2 det. = area sf)
• E.g ; T = [2 0 0][0 2 0][0 0 2] , T = enlargement sf 2 .'. determinant = 8 (as 2^3 = 8)
• - A minor determinant is the determinant of a smaller matrix within a larger matrix (e.g. the det. of a 2x2 inside a 3x3)
• - To find a minor determinant we delete a row and its corresponding column, and find the determinant of whats left
• - To calculate the determinant of M, where M = [a b c][d e f][g h i] , we do: |M| = a|e f||h i| - b|d f||g i| + c|d e||g h| , which then cancels to |M| = a(ei - fh) - b(di - fg) + c(dh - eg). Essentially, we find the 3 minor determinants (shown by |e f||h i|) , operate on them and use them to calculate the actual determinant of the 3x3.
• - Note the + - + pattern of the above , and if we start from a different point, the complete 3x3 pattern is: [+ - +][- + -][+ - +]
• You can find the determinant on your calculator , go to the matrix mode, define a 3x3 matrix, select options , matrix calc, options again, determinant.

Example Questions

• 1. Find the determinant of M , where M = [3 2 1][-1 0 4][5 -3 2]
• |M| = 3|0 4||-3 2| - 2|-1 4||5 2| + 1|-1 0||5 -3| , |M| = 3(0+12) - 2(-2-20) + 1(3-0) = 83 , so |M| = 83
• 2. The matrix [1 k 0][2 -3 k][1 k 2] is singular. Find k.
• As it's singular, |M| = 0 , so 0 = 1|-3 k||k 2| - k|2 k||1 2| + 0|2 -3||1 k| , 0 = 1(-6-k^2_ - k(4-k) + 0 , 0 = -6-k^2-4k+k^2 , 0 = -4k-6 , 4k=6 , k=-3/2 .

Notes on factorizing the determinant

• - If a row or column has a common multiple, it can be factorised out of the DETERMINANT (note this does not work for matrices, only det.)
• - e.g. : |a^2 2a ab||5 8 6b||1 0 b^3| = ab|a 2 1||5 8 6||1 0 b^2| = 2ab|a 1 1||5 4 6||1 0 b^2|
• - Adding or subtracting any row from any other row does not change the determinant (this also applies for columns)
• - When adding or subtracting , we should state what we are doing; use R for row and C for column , R1' = R1 + R2 would mean the new R1 is the old R1 added with R2.
• - Note a^2 - b^2 = (a-b)(a+b) , note a-b = -(b-a) , and note det(AB) = det(A)det(B)
• -e.g. : Factor out a-b for matrix |a^2 a 2b||b^2 b 2a||2 1 5| , R1' = R1-R2 : |a^2+b^2 a-b 2b-2a||b^2 b 2a||2 1 5| , same as: |(a-b)(a+b) a-b -2(a-b)||b^2 b 2a||2 1 5| , factor out (a-b) for: (a-b)|a+b 1 -2||b^2 b 2a||2 1 5|

Example questions

• 1. A = [3 a 1][a 5 4][4 a 0] , and B is a non singular matrix such that det(AB) = det(B) . Find the values of a.
• |A| = a^2 + 4a - 20 (work out) , det(AB) = det(A)det(B) , so det(A)det(B) = det(B) , we know B in non singular so det(B) ≠ 0 .'. det(A) = 1
• a^2 + 4a - 20 = 1 , a^2 + 4a -21 = 0 , a=-7 or a=3
• 2. Show |1 1 1||a b c||bc ca ab| can be written as 3 linear factors
• C1' = C1 - C2 : |0 1 1||a-b b c||bc-ca ca ab| which can be written: |0 1 1||a-b b c||-c(a-b) ca ab|
• Factorise (a-b) from C1: (a-b)|0 1 1||1 b c||-c ca ab|
• C2' = C2 - C3 : (a-b)|0 0 1||1 b-c c||-c ca-ab ab| which can be written: (a-b)|0 0 1||1 b-c c||-c -a(b-c) ab|
• Factorise (b-c) from C2: (a-b)(b-c)|0 0 1||1 1 c||-c -a ab|
• Now we expand for the determinant along R1 , and note there are 2 0's so we ignore the first 2 minor determinants , so we can say (a-b)(b-c)[1|1 1||-c -a|] , which is the same as (a-b)(b-c)(c-a). (-a+c = c-a)