# Question Video: Determining Whether Cramer’s Rule Is Useful for a System of Equations Whose Solution Set Is Infinite Mathematics • 10th Grade

Is Cramer’s rule useful for finding solutions to systems of linear equations in which there is an infinite set of solutions?

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### 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 And that’s where 𝐴 sub 𝑗 is the matrix 𝐴 whose 𝑗th column has been replaced by the constants in the matrix 𝐵.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is the determinant of the two-by-two matrix with elements 𝑎, 𝑒, 𝑏, 𝑓, where in this case it’s the second column that’s been replaced by 𝐵.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.