We're asked to find the cofactor matrix
of the given matrix.
Then find the transpose of the cofactor matrix,
which is the adjoint of A,
sometimes referred to as the adjugate of A.
If we take a look at our notes on the right,
the cofactor matrix consists of all the cofactors
from the original matrix.
And then the transpose of the cofactor matrix is equal
to the adjoint of matrix A.
Now remember, cofactor i,j is equal
to -1 raised to the power of i plus j
times the corresponding minor.
Let's work on determining the first row
in the cofactor matrix.
Cofactor A sub 1,1 is equal
to -1 raised to the power of 1 plus 1 times the minor.
To find the minor,
we eliminate row 1 column 1 of the original matrix,
which leaves us with a 2 by 2 matrix.
And, therefore, we multiply by the determinant
of this 2 by 2 matrix where row 1 is 4, zero,
and row 2 is zero, -2.
Simplifying, the square of -1 is +1,
the determinant is equal to 4 times -2,
which is -8, minus zero.
Simplifying, we have -8 for the entry in the cofactor matrix
for row 1 column 1.
Cofactor A sub 1, 2 is equal to -1
raised to the power of 1 plus 2, times the minor.
If we eliminate row 1 column 2,
the minor is the 2 by 2 determinant
where row 1 is -1, zero,
row 2 is 5, -2.
Simplifying, the cube of -1 is -1.
The determinant is -1 times -2,
which is 2,
minus zero times 5,
giving us 2 minus zero.
Simplifying, we have -2.
The entry in row 1 column 2 is -2.
Cofactor A sub 1, 3 is equal to -1
raised to the power of 1 plus 3,
times the minor.
If we eliminate row 1 column 3,
the minor is the 2 by 2 determinant
where row 1 is -1, 4,
row 2 is 5, zero.
Simplifying, the 4th power of -1 is 1.
The determinant is zero minus 20, or -20.
Simplifying, we have -20.
This is the entry for row 1 column 3.
And now let's work on row 2.
Cofactor A sub 1, 2 is equal to -1
raised to the power of 1 plus 2
times the corresponding minor.
If we eliminate row 2 column 1
from the original matrix,
we can see the minor is the determinant
where row 1 is 2, 3,
and row 2 is zero, -2.
The cube of -1 is -1.
The determinant is -4 minus zero.
Simplifying, we have +4.
This is the entry for row 2 column 1.
Cofactor A sub 2, 2 is equal to -1
raised to the power of 2 plus 2
times the corresponding minor.
If we eliminate row 2 column 2,
we have a 2 by 2 determinant
where row 1 is zero, 3,
row 2 is 5, -2.
Simplifying,
the 4th power of -1 is 1.
The determinant is equal to zero minus 15, or -15.
1 times -15 is -15,
the entry for row 2 column 2.
Cofactor A sub 2, 3 is equal to -1
raised to the power of 2 plus 3
times the corresponding minor.
If we eliminate row 2 column 3,
we have a 2 by 2 determinant
where row 1 is zero, 2,
row 2 is 5, zero.
Simplifying,
the 5th power of -1 is -1.
The determinant is zero minus 10, or -10.
Simplifying, we have +10.
This is the entry for row 2 column 3.
And now for the third row,
cofactor A sub 3, 1 is equal to -1
raised to the power of 3 plus 1
times the corresponding minor.
If we eliminate row 3 column 1,
the 2 by 2 determinant has a first row of 2, 3,
and a second row of 4, zero.
Simplifying, the 4th power of -1 is 1.
The determinant
is zero minus 12, or -12.
1 times -12 is -12,
the entry in row 3 column 1.
Cofactor A sub 3, 2
is equal to -1 raised to the power of 3 plus 2
times the minor.
If we eliminate row 3 column 2,
the minor is the 2 by 2 determinant
where row 1 is zero, 3,
and row 2 is -1, zero.
Simplifying,
the 5th power of -1 is -1.
The determinant is zero minus -3,
which simplifies to 3.
1 times 3 is -3,
the entry for row 3 column 2.
And then finally, cofactor A sub 3, 3
is equal to -1 raised to the power of 3 plus 3
times the minor.
If we eliminate row 3 column 3,
the minor is the 2 by 2 determinant
where row 1 is zero, 2,
and row 2 is -1, 4.
Simplifying, the 6th power of -1 is 1.
The determinant is zero minus -2,
which simplifies to 2.
1 times 2 is 2,
which is the entry in row 3 column 3.
And now that we have the cofactor matrix,
we can determine the adjoint of A
by finding the transpose of the cofactor matrix.
To transpose the cofactor matrix,
we make column 1 of the cofactor matrix
row 1 of the transpose,
we make column 2 of the cofactor matrix
row 2 of the transpose,
and we make column 3 of the cofactor matrix
row 3 of the transpose.
And this transpose is the adjoint of matrix A,
or the adjugate of A.
I hope you found this helpful.