Given matrix A,
we're asked to find the determinant of A;
Then determine if A is invertible;
Then find the cofactor matrix of A;
And finally, find the inverse of A.
Let's begin by finding the determinant.
We will perform cofactor expansion on row 2
because row 2 has 2 zero entries.
So starting with the first entry in row 2,
the determinant of matrix A is equal to zero
times cofactor 2, 1.
Because we have a zero in the product,
we don't have to find cofactor 2, 1.
So we move along to the second entry in row 2,
which is -2.
So we have plus -2 times cofactor 2, 1
which is equal to -1 raised to the power of 2 plus 2.
The exponent is 2 plus 2
because the element is in row 2, column 2.
And then we have times the minor.
The minor is equal to the determinant of the matrix formed
after eliminating row 2 and column 2 from the matrix,
which is the row and column of -2.
This leaves us with a 2 by 2 determinant
where row 1 is 2, -4
and row 2 is 4, 6
which we see here.
And now I move along to the third entry in row 2,
which is another zero,
which gives us plus zero times cofactor of 3, 1.
Again, because we have a factor of zero,
we could stop, the product is zero.
So simplifying,
the first product and third product are both zero.
Simplifying, we have -2 times,
the 4th power of -1 is equal to +1.
And the determinant is equal to 2 times 6, which is 12,
minus -4 times 4,
which is -16.
Simplifying,
12 minus -16 is equal to 12 plus 16, or 28.
The determinant is equal to -2 times 28,
which is equal to -56.
Number 2 is A invertible
and since the determinant does not equal zero,
matrix A is invertible,
which means matrix A does have an inverse.
For number 3,
we're asked to find the cofactor matrix.
Because we have a 3 by 3 matrix,
the cofactor matrix is defined
here in the upper right-hand corner.
Notice how we have to find 9 cofactors.
And I've already set all this up to save some time.
Let's go through the first row.
Cofactor of 1, 1 is equal to -1,
raised to the power of 1 plus 1
times the corresponding minor.
The minor is equal to the determinant of the matrix
after eliminating row 1 and column 1 from the matrix.
If we eliminate row 1 and column 1,
we can see we have a 2 by 2 matrix.
The minor is equal to the determinant of this 2 by 2 matrix,
where row 1 is -2, zero
and row 2 is -2, 6.
The next entry in the cofactor matrix is cofactor 1, 2
which is equal to -1 raised to the power of 1 plus 2
times the corresponding minor.
The minor's equal to the determinant of the matrix
after eliminating row 1 and now column 2.
We have a 2 by 2 determinant
where row 1 is zero, zero
and row 2 is 4, 6.
The entry in row 1, column 3 is cofactor of 1, 3
which is equal to -1 raised to the power of 1 plus 3
times the corresponding minor.
To find the minor, we eliminate row 1, column 3.
The minor is equal to the determinant
of the remaining 2 by 2 determinant
where row 1 is zero, -2
and row 2 is 4, -2.
Now I'm going to go and stop here,
but the remaining cofactors are already listed
and then we need to simplify them
to find the cofactor matrix.
Again, let's at least simplify the first row.
In row 1, column 1,
we have the square of -1,
which is +1,
and the determinant is equal to -12 minus zero, or -12.
1 times -12 gives us -12
in row 1, column 1 of the cofactor matrix.
For the entry in row 1, column 2
we got the cube of -1, which is -1.
And the determinant is zero minus zero,
which, of course, is zero.
Giving us an entry of zero in row 1, column 2
of the cofactor matrix.
Let's go and take a look at one more.
In row 1, column 3
we have the 4th power of -1,
which is +1.
And the determinant is zero minus -8,
which simplifies to 1 times +8, which is 8,
giving us an entry of 8 in row 1, column 3
of the cofactor matrix.
So you may want to pause the video
and verify the remaining entries for the cofactor matrix.
For number 4,
we're asked to find the inverse of matrix A
and we'll do this using the adjoint method
where A inverse is equal to 1 divided by the determinant
of matrix A
times the adjoint of A,
where adjoint A is equal to the transpose
of the cofactor matrix.
So before we find the inverse matrix,
we do have to find the adjoint of A
by transposing the cofactor matrix that we just found.
So we just found this cofactor matrix and, therefore,
the adjoint of A is equal to the transpose of this matrix.
To transpose the matrix,
the first column of the cofactor matrix becomes
the first row in the transpose
or the first row of the adjoint of A.
The second column of the cofactor matrix becomes
the second row of the transpose
or of the adjoint of A.
And the third column of the cofactor matrix becomes
the third row of the transpose,
or the third row of the adjoint matrix.
So now that we have the adjoint of A
and we already have the determinant,
we can use the formula to find A inverse.
Let's do this on the next slide.
The determinant of A is equal to -56,
giving us 1 divided by -56.
And then we have times the adjoint of A,
which, again, is the transpose of the cofactor matrix.
Next we perform the scalar multiplication,
and then simplify the fractions.
Simplifying the first row of A inverse
is 3/14, 2/7, and 1/7.
The second row is zero, - 1/2, zero.
And the third row is - 1/7, - 5/14, and 1/14.
I hope you found this helpful.