### Video Transcript

Consider the following linear
system of equations in matrix form: the matrix zero, negative three, seven, zero,
negative one, one, one, one, one multiplied by the matrix 𝑥, 𝑦, 𝑧 equals the
matrix one, six, one. Write the determinant Δ 𝑧.

To start off with, we should make
sure we understand the matrix interpretation of the linear system of equations given
to us. We can see that the system has
three unknowns: 𝑥, 𝑦, and 𝑧, because these are the only variables found in the
variable matrix. To the left of this is the matrix
of coefficients. We call it this because if we
multiplied this system out, these numbers would form the coefficients of the 𝑥-,
𝑦-, and 𝑧-variables. Lastly, we have the constant
matrix, which forms the right-hand side of the system of equations.

Now, we are being asked to find the
determinant Δ 𝑧. Recall that this determinant is
used during the calculation of Cramer’s rule, and is formed by using parts of the
coefficient matrix and the constant matrix. We should also note that we have
been asked to write the determinant, but not to evaluate it. So, our answer should be a
three-by-three determinant rather than a number. Let us recall the process for
finding Δ 𝑥, Δ 𝑦, or Δ 𝑧.

We can start off by writing out the
determinant of the coefficient matrix, which is just called Δ. This is just the matrix on the left
of the system, but rewritten using bars instead of brackets. The next step is to label the
columns using the unknowns, going in order from left to right and top to bottom. In this case, this means the
unknowns 𝑥, 𝑦, and 𝑧. Lastly, we have to replace the 𝑥,
𝑦, or 𝑧 column with values from the constant matrix to form Δ 𝑥, Δ 𝑦, or Δ
𝑧. Now, the constant matrix refers to
the matrix on the right-hand side of the system. Also, we are only interested in Δ
𝑧 in this question, so we will be looking to replace the 𝑧 column.

Let us see how these steps can be
applied to this example. For the first step, we write out Δ,
which is the coefficient matrix highlighted above, but written as a determinant. Next, going from left to right, we
assign each of the three columns of the determinant to the variables 𝑥, 𝑦, or 𝑧
in the same order they appear in the variable matrix.

Finally, since we want to find Δ
𝑧, we look to replace the 𝑧-column, which is the third column along, with the
constant matrix. This means we want to replace the
column seven, one, one, with one, six, one. The rest of the entries remain the
same. This gets us the determinant, zero,
negative three, one, zero, negative one, six, one, one, one. As mentioned previously, we do not
need to explicitly evaluate this determinant for this question. Thus, this is our final answer for
Δ 𝑧.