find the minors and cofactors of all the elements in the following matrix
and hence, write down the cofactor matrix of A
The minor of the 12 is the 2×2 determinant which
contains only the elements of A that are NOT
in the same row or the same column as 12.
The minor of the 12 is this determinant:
|8 3|
|7 0| = 8·0 - 3·7 = 0 - 21 = -21
To find the cofactor, we multiply this minor -21,
by either +1 or -1. To decide which, we notice
that 12 is in row 1 and column 1. We add 1+1 and
get 2, and since 2 is an even number, we multiply
the minor -21 by +1 and get -21, so,
the cofactor of 12 is -21.
---------------------------------
The minor of the 7 in the top row is the 2×2
determinant which contains only the elements
of A that are NOT in the same row or the same
column as that 7.
The minor of that 7 is this determinant:
|5 3|
|6 0| = 5·0 - 3·6 = 0 - 18 = -18
To find the cofactor, we multiply this minor -18,
by either +1 or -1. To decide which, we notice
that that 7 is in row 1 and column 2. We add 1+2 and
get 3, and since 3 is an odd number, we multiply
the minor -18 by -1 and get +18, so,
the cofactor of the 7 in the top row is 18.
------------------------------------------------
The minor of the 0 in the top row is the 2×2
determinant which contains only the elements
of A that are NOT in the same row or the same
column as that 0.
The minor of that 0 is this determinant:
|5 8|
|6 7| = 5·7 - 8·6 = 35 - 48 = -13
To find the cofactor, we multiply this minor -18,
by either +1 or -1. To decide which, we notice
that that 0 is in row 1 and column 3. We add 1+3 and
get 4, and since 4 is an even number, we multiply
the minor -13 by +1 and get -13, so,
the cofactor of the 0 in the top row is -13.
---------------------------------
The minor of the 5 is the 2×2 determinant which
contains only the elements of A that are NOT
in the same row or the same column as 5.
The minor of 5 is this determinant:
|7 0|
|7 0| = 7·0 - 0·7 = 0 - 0 = 0
To find the cofactor, we multiply this minor 0,
by either +1 or -1. To decide which, we notice
that 5 is in row 2 and column 1. We add 2+1 and
get 3, and since 2 is an odd number, we multiply
the minor 0 by -1 and get 0, so,
the cofactor of 5 is 0.
[Yes, I know that since the minor of 5 is 0, that
whether we multiplied it by +1 or -1 it would still
be 0, but I was just following through the process
as if it had not been 0]
---------------------------------
The minor of the 8 is the 2×2 determinant which
contains only the elements of A that are NOT
in the same row or the same column as 8.
The minor of 8 is this determinant:
|12 0|
| 6 0| = 12·0 - 0·6 = 0 - 0 = 0
To find the cofactor, we multiply this minor 0,
by either +1 or -1. To decide which, we notice
that 8 is in row 2 and column 2. We add 2+2 and
get 4, and since 4 is an even number, we multiply
the minor 0 by +1 and get 0, so,
the cofactor of 8 is 0.
---------------------------------
The minor of the 3 is the 2×2 determinant which
contains only the elements of A that are NOT
in the same row or the same column as the 3.
The minor of 3 is this determinant:
|12 7|
| 6 7| = 12·7 - 7·6 = 84 - 42 = 42
To find the cofactor, we multiply this minor 42,
by either +1 or -1. To decide which, we notice
that 3 is in row 2 and column 3. We add 2+3 and
get 5, and since 5 is an odd number, we multiply
the minor 42 by -1 and get -42, so,
the cofactor of 3 is -42.
---------------------------------
The minor of the 6 is the 2×2 determinant which
contains only the elements of A that are NOT
in the same row or the same column as the 6.
The minor of 6 is this determinant:
|7 0|
|8 3| = 7·3 - 0·8 = 21 - 0 = 21
To find the cofactor, we multiply this minor 21,
by either +1 or -1. To decide which, we notice
that 6 is in row 3 and column 1. We add 3+1 and
get 4, and since 4 is an even number, we multiply
the minor 21 by +1 and get 21, so,
the cofactor of 6 is 21.
----------------------------------
The minor of the 7 in the bottom row is the 2×2
determinant which contains only the elements
of A that are NOT in the same row or the same
column as that 7.
The minor of that 7 is this determinant:
|12 0|
| 5 3| = 12·3 - 0·5 = 36 - 0 = 36
To find the cofactor, we multiply this minor 36,
by either +1 or -1. To decide which, we notice
that that 7 is in row 3 and column 2. We add 3+2 and
get 5, and since 5 is an odd number, we multiply
the minor 36 by -1 and get -36, so,
the cofactor of the 7 in the bottom row is -36.
------------------------------------------------
The minor of the 0 in the bottom row is the 2×2
determinant which contains only the elements
of A that are NOT in the same row or the same
column as that 0.
The minor of that 0 is this determinant:
|12 7|
| 5 8| = 12·8 - 7·5 = 96 - 35 = 61
To find the cofactor, we multiply this minor 61,
by either +1 or -1. To decide which, we notice
that that 0 is in row 3 and column 3. We add 3+3 and
get 6, and since 6 is an even number, we multiply
the minor 61 by +1 and get +61, so,
the cofactor of the 0 in the bottom row is 61.
------------------------------------------------
So the cofactor matrix of the matrix
is this matrix
Edwin