Matrices cofactor expansion
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Matrices cofactor expansion

[From: ] [author: ] [Date: 12-08-27] [Hit: ]
.could someone please confirm?my answer is 40..........
This is a bit of a strange question, but i think i got the answer...could someone please confirm? or say if its wrong

ok so

Calculate the second term in the cofactor expansion along the third row for the matrix

3 -5 2 1
-3 4 1 -2
4 -3 -2 1
-5 3 2 1

my answer is 40....or maybe its -40....?

please help

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That can't be right. the [3][2] coefficient is -3, so the result has to be a multiple of 3.

Start with the sign. Row 3, column 2: 3+2 is odd so it's minus. Take (-3) as the coefficient and strike out the 3rd row and 2nd column to leave
| 3 2 . 1 |
|-3 1 -2 |
|-5 2 . 1|

So, the whole determinant is:

- (-3) [(3)(1)(1) + (2)(-2)(-5) + (1)(-3)(2) - (1)(1)(-5) - (2)(-3)(1) - (3)(-2)(2)]
= 3*(3 + 20 - 6 + 5 + 6 + 12) = 120

It looks like you got the right answer for just the reduced determinant, so maybe I don't understand the question. But 120 will be the term that adds to produce the full determinant.

-
This term will be the following:

(-1)^(2+3) (-3) |3 2 1|
..........................|-3 1 -2|
..........................|-5 2 1|,

which, after using minors to compute this 3 by 3 determinant is just (-1)(-3)(40) = 120.

What seems apparent is that you only consider the determinant of the minor matrix for this term and not the other factors needed to make a cofactor matrix. Remember that if M_(i,j) is the minor associated to the (i,j)-th term of an n by n matrix, then the cofactor of for the (i,j_th term is C_(i,j) = (-1)^(i+j) a_(i,j) M_(i,j), where a_(i,j) is the (i,t)-th term discussed.
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