We're asked to find
the minors and the cofactors of the given 2 by 2 matrix.
The minor i, j is equal to the determinant
of the matrix formed by eliminating row i and column j
from the original matrix.
So, for example, minor 1, 1
is equal to the determinant of the matrix formed
by eliminating row 1 and column 1 from the given matrix.
If we eliminate row 1 and column 1 from the given matrix,
notice how we have a 1 by 1 matrix with an entry of -1.
But the minor of 1, 1
is equal to the determinant
of the 1 by 1 matrix with an entry of -1.
Whenever we have a 1 by 1 matrix,
if I have the determinant, is the entry itself,
and therefore the determinant
in minor 1, 1 is equal to -1.
Minor 1, 2 is equal to the determinant of the matrix
after eliminating row 1 and column 2.
Minor 1, 2 is equal to the determinant of the 1 by 1 matrix
with an entry of 2,
which is equal to positive 2.
Minor 2, 1 is equal to the determinant of the matrix formed
after eliminating row 2 column 1.
Minor 2, 1 is equal to the determinant
of the 1 by 1 matrix with an entry of -6,
which is equal to -6.
Minor 2, 2 is equal to the determinant of the matrix formed
after eliminating row 2 and column 2.
Minor 2, 2 is equal to the determinant
of the 1 by 1 matrix with an entry of 5,
which is equal to 5.
I purposely did not use vertical bars
to represent the determinant
because it's a little bit confusing.
If we use vertical bars to represent a determinant
of a 1 by 1 matrix,
they might be confused with absolute value.
For example,
for the determinant of a 1 by 1 matrix
where the entry is -6,
we could use vertical bars like this
to represent the determinant of the 1 by 1 matrix.
But again,
this might be confused with the absolute value of -6.
When referring to determinant,
the determinant of the 1 by 1 matrix,
with an entry of -6 is -6.
But again, because that can be confusing,
I am using different notation for determinant.
And now let's find the cofactors of the matrix.
Cofactor A sub i, j
sometimes referred to as just C sub i, j
is equal to -1 raised to the power of i plus j
times the corresponding minor.
So cofactor of 1, 1 is equal to -1
raised to the power of 1 plus 1
times minor 1, 1 which we already found.
And -1 raised to the 2nd power is +1,
giving us +1 times -1.
Cofactor of 1, 1 is equal to -1.
Cofactor of 1, 2 is equal to -1 raised to the power
of 1 plus 2 times minor 1, 2
which is equal to,
well, the cube of -1 is -1
and we know minor 1, 2 is 2.
1 times 2 is equal to -2.
Next we have cofactor 2, 1
which is equal to -1 raised to the power of 2 plus 1
times minor 2, 1.
Again, we have the cube of -1, which is -1
and we know minor 2, 1 is -6.
1 times -6 is +6.
And finally, cofactor of 2, 2 is equal
to -1 raised to the power of 2 plus 2
times minor 2, 2.
Which is equal to the 4th power of -1,
which is +1 times,
minor 2, 2 is 5.
1 times 5 is 5.
Before we go, I do want to mention,
if we take a look at the powers of -1,
we should be able to recognize,
when the exponent is even,
the result is +1.
When the exponent is odd, the result is -1.
And sometimes you'll see this represented
in the form of a matrix as we see here on the left in red.
Where if the sum of the row and column is even,
the result is plus for +1.
And if the sum of the row and column is odd,
the result is -1 represented by the negative.
I hope you found this helpful.