### Video Transcript

Given that the product of a
three-by-three matrix with elements negative one, negative six, negative six, zero,
negative two, negative four, two, four, and seven and a vector containing elements
𝑥, 𝑦, and 𝑧 is equal to the vector containing elements negative eight, two, and
negative nine, find the values of 𝑥, 𝑦, and 𝑧.

We know that if matrix 𝐴 is
multiplied by vector 𝐗 and it equals vector 𝐁, that vector 𝐗 equals the inverse
matrix of 𝐴 multiplied by vector 𝐁. We should also remember that order
matters. We cannot multiply 𝐁 by the
inverse matrix of 𝐴. We must multiply it in this order,
the inverse of 𝐴 times 𝐁. This means that our first step is
to find the inverse of matrix 𝐴.

You could use a graphing calculator
to help you do this step. But we will take a look at how to
do the steps by hand without a graphing calculator. This is a multistep process and
will be the most complex part of solving this problem.

The first step in finding the
inverse of matrix 𝐴, if the inverse exists, is to find the cofactor matrix, 𝐶,
equal to the three-by-three matrix 𝐶 sub 𝑖𝑗, whose entries are the determinants
of the corresponding matrix minors multiplied by the alternating sign, negative one,
to the power of 𝑖 plus 𝑗. Then, we transpose the cofactor
matrix to find the adjugate matrix, sometimes referred to as the classical adjoint
matrix. Then, we will calculate the
determinant of 𝐴 and multiply the adjugate matrix by the reciprocal of the
determinant. Since 𝐴 is a three-by-three
matrix, we must use the following formula to find the three-by-three cofactor
matrix.

Notice the pattern of alternating
positive and negative signs in front of each determinant. If we assign the wrong sign to one
element, it can completely change our final result.

To begin, we find the determinant
in the first row, first column. We look to matrix 𝐴 to find the
four elements in the 𝑒-, 𝑓-, ℎ-, and 𝑖-positions. These are negative two, negative
four, four, and seven. To calculate the determinant, we
multiply the top-left element by the bottom-right element. And then we subtract the
bottom-left element multiplied by the top-right element. And so the first element of our
cofactor matrix is found by negative two times seven minus four times negative four,
which is negative 14 minus negative 16, which equals positive two. So the top-left element in our new
matrix is going to be positive two.

The determinant from the first row,
second column would then look like this: zero times seven minus two times negative
four, which is zero minus negative eight, which equals positive eight. Notice, however, that because of
its position in the cofactor matrix, we need to apply a negative sign. So that element in the first row,
second column is negative eight.

Now we’re looking for the element
in the first row, third column: zero times four minus two times negative two. This element comes out to positive
four. Let’s repeat this process for the
second row of cofactors. The signs for this row are
negative, positive, negative. The first determinant in this row
comes out to negative 18. But we need to take the negative of
negative 18. And so we have positive 18 in the
second row, first column position.

In the second row, second column,
we have positive five. And the determinant in the second
row, third column comes out to positive eight. But we need to take the negative of
that value. So this element will actually be
negative eight.

And finally we’ll calculate the
determinants in the third row using first a positive sign, then negative, then
positive again. For the third row, first column, we
get positive 12. In the third row, second column, we
get four. But we need to multiply that by
negative one, which gives us negative four. And in the final position, we get
positive two.

We now have our complete cofactor
matrix. Our next step is to find the
adjugate matrix. We do this by transposing the rows
and the columns. And that looks like a reflection
across the diagonal. The elements along the diagonal
remain the same. But we’ll switch the negative eight
and positive 18, the four with the 12, and the negative eight with the negative
four. So the adjugate of matrix 𝐴 is
two, 18, 12, negative eight, five, negative four, four, negative eight, two.

Our final step is to then multiply
the adjugate matrix by the reciprocal of the determinant of the original matrix
𝐴. To calculate the determinant, we
can use this formula. We’ll multiply each element in the
top row by the matching cofactors we found in the previous step. The determinant of the first row,
first column was two. The first row, second column was
eight. And the first row, third column was
four. The expression simplifies to
negative two plus 48 minus 24. So the determinant of 𝐴 is 22. So the reciprocal of the
determinant of matrix 𝐴 is then one over 22. Altogether, the inverse of matrix
𝐴 is one over 22 times the adjugate.

After multiplying each element by
one over 22, the product looks like this. As we originally said, to solve
this system, we need to calculate the inverse of matrix 𝐴 times 𝐁. The resulting array will contain
three elements: a value for 𝑥, a value for 𝑦, and a value for 𝑧.

To find 𝑥, we multiply negative
eight by one over 11, two by nine over 11, and negative nine by six over 11, then
find the sum of these products. This expression simplifies to
negative 44 over 11. So the value of 𝑥 is negative
four.

The value of 𝑦 is found by
calculating negative eight times negative four over 11 plus two times five over 22
plus negative nine times negative two over 11, which simplifies to 55 over 11. So 𝑦 is equal to positive
five.

And finally to solve for 𝑧, we
must calculate negative eight times two over 11 plus two times negative four over 11
plus negative nine times one over 11. This expression simplifies to
negative 33 over 11. So 𝑧 is equal to negative
three.

Therefore, under these conditions,
𝑥 is equal to negative four, 𝑦 is equal to five, and 𝑧 is equal to negative
three.