Video Transcript
Noah is solving simultaneous
equations using Cramer’s rule. He writes down the following. Δ sub 𝑥 is the determinant of the
three-by-three matrix one, two, three, four, negative three, negative two, two, one,
negative four. Δ sub 𝑦 is the determinant of the
three-by-three matrix two, one, three, negative three, four, negative two, negative
one, two, negative four. Δ sub 𝑧 is the determinant of the
three-by-three matrix two, two, one, negative three, negative three, four, negative
one, one, two. What does he write down for Δ?
In this question, we are told that
Noah is solving a system of equations by using Cramer’s rule. However, we are not given the
system of equations. Instead, we are given the
expressions used to calculate Δ sub 𝑥, Δ sub 𝑦, and Δ sub 𝑧. We need to use these expressions to
determine the expression needed to calculate Δ.
To answer this question, we need to
start by recalling how we can use Cramer’s rule to solve a system of equations. Since the expressions we are given
are determinants of three-by-three matrices, we can recall the version for a system
of three linear equations. 𝑎𝑥 plus 𝑏𝑦 plus 𝑐𝑧 equals 𝑗;
𝑑𝑥 plus 𝑒𝑦 plus 𝑓𝑧 equals 𝑘; 𝑔𝑥 plus ℎ𝑦 plus 𝑖𝑧 equals 𝑙.
Cramer’s rule tells us that if the
determinant of the matrix of coefficients is nonzero — that is, Δ equals the
determinant of the three-by-three matrix 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ, 𝑖 is
nonzero — then the system has a unique solution given by 𝑥 equals Δ sub 𝑥 over Δ,
𝑦 equals Δ sub 𝑦 over Δ, and 𝑧 equals Δ sub 𝑧 over Δ, where Δ sub 𝑥, 𝑦, and 𝑧
are found by replacing the column for that variable in the expansion of Δ with the
constants from the simultaneous equations. For instance, we can find Δ sub 𝑥
by replacing the first column in the expression for Δ with the entries 𝑗, 𝑘, and
𝑙 to get the following expression for Δ sub 𝑥.
We want to find an expression for
Δ. So we want to find the values of
these nine constants. We can find six of these constants
by using our expression for Δ sub 𝑥. Since the second and third column
in the expressions for Δ sub 𝑥 and Δ are the same, we can add these two columns
into our expression for Δ. We can apply this same reasoning to
the other two expressions we have. We see that the first column of Δ
sub 𝑦 and Δ sub 𝑧 tells us the values of 𝑎, 𝑑, and 𝑔. We can then add this column into
the expression for Δ and take the determinant to get that Δ must be equal to the
determinant of the three-by-three matrix two, two, three, negative three, negative
three, negative two, negative one, one, negative four.
