Determinants

Question 1

Cramer’s rule

Answer 1

Sure, here’s an example of solving a system of linear equations using Cramer’s Rule:

Suppose we have the following system of two linear equations in two variables:

x + 2y = 5
3x - 4y = 2

To solve for x and y using Cramer’s Rule, we first find the determinant of the coefficient matrix A:

A = [1 2; 3 -4]

|A| = (1)(-4) - (2)(3) = -10

Next, we find the determinant of the matrix obtained by replacing the first column of A with the constants on the right-hand side of the equations:

A1 = [5 2; 3 -4]

|A1| = (5)(-4) - (2)(3) = -26

Then, we find the determinant of the matrix obtained by replacing the second column of A with the constants on the right-hand side of the equations:

A2 = [1 5; 3 2]

|A2| = (1)(2) - (5)(3) = -13

Finally, we can find the values of x and y by dividing the determinant of each modified matrix by the determinant of A:

x = |A1|/|A| = (-26)/(-10) = 13/5

y = |A2|/|A| = (-13)/(-10) = 1.3

Therefore, the solution to the system of linear equations is x = 13/5 and y = 1.3.