Video Transcript
Is Cramer’s rule useful for finding
solutions to systems of linear equations in which there is an infinite set of
solutions?
To answer this question, we recall
that for a system of 𝑛 linear equations in 𝑛 unknowns, where 𝐴 is the 𝑛-by-𝑛
coefficient matrix, 𝑥 is the matrix of unknowns, and 𝐵 is the matrix of constants,
if the determinant of the matrix 𝐴 is not equal to zero, then the unique solution
for the unknowns 𝑥 one, 𝑥 two, up to 𝑥 𝑛 is given by 𝑥 sub 𝑗 is equal to the
determinant of the matrix 𝐴 sub 𝑗 divided by the determinant of matrix 𝐴. And that’s for 𝑗 is equal to one
to 𝑛. And that’s where 𝐴 sub 𝑗 is the
matrix 𝐴 whose 𝑗th column has been replaced by the constants in the matrix 𝐵. So, for example, for a system of
two linear equations with two unknowns, expressed in matrix form, this is a
two-by-two matrix multiplied by a column matrix 𝑥, 𝑦 is equal to the constants
matrix 𝑒, 𝑓.
Then, Cramer’s rule in this
two-by-two case gives us the solutions 𝑥 is equal to Δ𝑥 divided by Δ and 𝑦 is
equal to Δ𝑦 divided by Δ. And that’s where Δ𝑥 is the
determinant of the two-by-two matrix with elements 𝑒, 𝑏, 𝑓, 𝑑. And this is the determinant of the
matrix 𝐴 with its first column replaced by the elements of 𝐵. Similarly, Δ𝑦 is the determinant
of the two-by-two matrix with elements 𝑎, 𝑒, 𝑏, 𝑓, where in this case it’s the
second column that’s been replaced by 𝐵. And Δ is the determinant of the
matrix 𝐴.
Now, making some space and going
back to our question, we want to know, is Cramer’s rule useful for finding solutions
to systems of linear equations where the solution set is infinite? And we can answer this question by
noting that there are two ways a linear system can have an infinite number of
solutions. The first is if there are more
variables than equations. For example, the system shown has
three unknowns — that’s 𝑥, 𝑦, and 𝑧 — but only two equations. In such cases, the coefficient
matrix 𝐴 is not square, and this is a necessary condition to apply Cramer’s
rule. And our second way in which a
linear system can have an infinite number of solutions is if the determinant of the
coefficient matrix 𝐴 is equal to zero. In this case, we can’t apply
Cramer’s rule since in the formula we would be dividing by zero — that’s the
determinant of 𝐴 — and this we cannot do.
So, in either of these cases, we
cannot apply Cramer’s rule. And so we must conclude that
Cramer’s rule would not be useful for finding the solutions to a system of equations
in which there’s an infinite set of solutions. And so our answer is therefore
no.
