We're asked to find the minors and cofactors
of the given 3 by 3 matrix.
Let's begin by determining the minors.
Minor i, j is equal to the determinant
of the matrix formed by eliminating row i and column j.
So minor 1, 1 is equal to the determinant
of the matrix formed by eliminating row 1 and column 1.
Notice how this leaves us with a 2 by 2 matrix.
M, 1 is equal to the 2 by 2 determinant
where the first row
is 4 and 5
and the second row is -6, 9.
Let's go through and find all the 2 by 2 determinants
and then we'll come back and evaluate them.
To find minor 1, 2
we eliminate row 1 and column 2,
which gives us a 2 by 2 determinant
where the first row is -2, 5
and the second row is zero, 9.
To find minor 1, 3
we eliminate row 1 column 3.
We are left with the 2 by 2 determinant
where the first row is -2, 4
and the second row is zero, -6.
To find minor 2, 1
we now eliminate row 2 column 1.
Again we have a 2 by 2 determinant
where the first row is 3, 2
and the second row is -6, 9.
To find minor 2, 2
we eliminate row 2 column 2.
And again we have a 2 by 2 determinant
where the first row is 1 and 2
and the second row is zero, 9.
For minor 2, 3
we eliminate row 2 column 3.
The first row is 1, 3.
The second row is zero, -6.
For minor 3, 1
we eliminate row 3 column 1.
The first row of the determinant is 3, 2
and the second row is 4, 5.
For minor 3, 2
we eliminate row 3 column 2.
The first row is 1, 2
and the second row is -2, 5.
And finally, for minor 3, 3
we eliminate row 3 column 3.
Row 1 is 1, 3.
Row 2 is -2, 4.
And now we'll go ahead and evaluate the determinants.
Because we have 2 by 2 determinants,
we find the product of the main diagonal
and subtract the product of the minor diagonal.
For minor 1, 1
we have 36 minus -30,
which simplifies to 36 plus 30, or 66.
For minor 1, 2
we have -18 minus zero, which is -18.
Fr minor 1, 3
we have 12 minus zero,
which is 12.
For minor 2, 1
we have 27 minus -12,
which is equal to 27 plus 12, or 39.
For minor 2, 2
we have 9 minus zero, which is 9.
For minor 2, 3
we have -6 minus zero, which is -6.
For minor 3, 1
we have 15 minus 8,
which is 7.
For minor 3, 2
we have 5 minus -4,
which simplifies to 5 plus 4, or 9.
And finally,
minor 3, 3 is 4 minus -6,
which simplifies to 4 plus 6, which is 10.
Now that we have all the minors,
let's find the cofactors.
Cofactor i, j is equal to -1
raised to the power of i plus j times the minor i, j.
So cofactor of 1,1 is equal to -1
raised to the power of 1 plus 1
times minor 1, 1.
Well, the square of -1 is +1
and minor 1, 1 is 69.
1 times 69 is 69.
The cofactor of 1, 2 is equal
to -1 raised to the power of 1 plus 2
times minor 1, 2.
The cube of -1 is -1
and minor 1, 2 is -18.
1 times -18 is +18.
The cofactor of 1, 3 is equal to -1
raised to the power of 1 plus 3
times minor 1, 3.
The 4th power of -1 is +1.
The minor of 1, 3 is 12.
1 times 12 is 12.
Cofactor of 2, 1 is equal to -1
raised to the power of 2 plus 1
times minor 2, 1.
The cube of -1 is -1.
Minor 2, 1 is 39.
1 times 39 is -39.
Cofactor of 2, 2 is equal to -1
raised to the power of 2 plus 2
times minor 2, 2.
Which gives us +1 times 9, which is 9.
Cofactor of 2, 3
is equal to -1 raised to the power of 2 plus 3
times minor 2, 3.
The 5th power of -1 is -1.
Minor 2, 3 is -6.
1 times -6 is +6.
The cofactor of 3, 1 is equal to -1
raised to the power of 3 plus 1
times minor 3, 1
which is equal to the 4th power of -1, which is 1,
and minor of 3, 1 is 7
and 1 times 7 to 7.
Cofactor of 3, 2
is equal to -1 raised to the power of 3 plus 2
times minor 3, 2
which gives us -1 times 9, which is -9.
And finally, the cofactor of 3, 3
is equal to -1 raised to the power of 3 plus 3
times minor 3, 3
which is 1 times 10, which is 10.
I hope you found this helpful.