# Question Video: Using Cramer’s Rule Mathematics • 10th Grade

Consider the following system of equations in matrix form: [2, 2, 3 and 1, −1, −3 and 0, 1, 0][𝑥, 𝑦, 𝑧] = [2, 1, 2]. Write the determinant Δ𝑦.

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### Video Transcript

Consider the following system of equations in matrix form. The matrix two, two, three, one, negative one, negative three, zero, one, zero times the matrix 𝑥, 𝑦, 𝑧 equals the matrix two, one, two. Write the determinant Δ𝑦.

Recall that Cramer’s rule is a method for solving systems of equations using the determinants of certain matrices. In order to apply Cramer’s rule, we need to calculate the determinants Δ𝑥, Δ𝑦, and Δ𝑧. The determinant Δ𝑥 is the determinant of the matrix obtained by replacing the column of matrix 𝐴 corresponding to 𝑥 with the matrix 𝐵. This is the column corresponding to 𝑥. The determinant Δ𝑦 is the determinant of the matrix obtained by replacing the column corresponding to 𝑦 in matrix 𝐴 with the matrix 𝐵.

Thus, Δ𝑦 is the determinant of the matrix two, two, three, one, one, negative three, zero, two, zero. Note that the question doesn’t actually ask us to evaluate this determinant, but you can pause the video now and have a go. Its value is 18.