SOLUTION: What is the determinant of the matrix F and G F= 0 3 0 0 0 a 2 b c d e 1 f g h i 0 j k l m 0 n o p and G = a b

Question 1207027: What is the determinant of the matrix F and G
F=
0 3 0 0 0
a 2 b c d
e 1 f g h
i 0 j k l
m 0 n o p
and
G =
a b c d
e-a f-b g-c h-d
i-m j-n k-o l-p
m n o p
if I have matrix
M=
a b c d
e f j h
i j k l
m n o p
that has a det=|M|=5
---------------------------------
so with matrix F I took 3 as common factor and multiplied it by |M|=5 which means |F|=15
and with G I just transform it to M which would mean |G|=|M|=5
is this correct?
any help is highly appreciated

Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.

According to the cofactoring theorem for matrices and determinants,
you should take common factor -3 (= negative 3) with matrix F
and multiply it by |M| = 5, so the final determinant for matrix F is

                |F| = (-3)*5 = -15.

With matrix G, your understanding is correct and the final answer is correct, too.

----------------

To learn on how cofactoring works for 3x3 determinants, see the lessons
    - Determinant of a 3x3 matrix
    - Co-factoring the determinant of a 3x3 matrix
in this site.   It is very similar for matrices of higher dimensions.

Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Given matrices

We are told that |M| = 5
i.e. the determinant of matrix M is 5.

--------------------------------------------------------------------------

In the 1st row of matrix F, there are almost all zeros here.
This means it is the best choice when doing a cofactor expansion.

Circle the nonzero element of this row.

This circled element is in the 1st row and 2nd column.

Anything in this row/column combo that isn't circled is then crossed off.
I denote this as a red X

Erase those red X's.
The resulting 4x4 submatrix (temporarily ignoring the "3") is matrix M.
We will multiply 3 by this matrix determinant and stick a negative 1 out front. Recall the plus minus signs alternate like a checkerboard.

The spacing is a bit strange. I apologize. I couldn't figure out how to get it lined up perfectly.
Use the minus sign in row1,column2.

So this is why |F| = -3*|M| = -3*5 = -15

--------------------------------------------------------------------------

Refer to Theorem 3.2.4 of this link
https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/03%3A_Determinants/3.02%3A_Properties_of_Determinants

Due to that theorem, |G| = |M| = 5

This theorem states that adding a multiple of one row to another row does not change the determinant.
Part 4 of Theorem 3.2.8 rephrases this theorem.

--------------------------------------------------------------------------

Summary
|F| = -15
|G| = 5

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