Cramer's Rule Application

Cramer's rule solves a linear system by expressing each variable as a ratio of determinants, the determinant of a column-replaced matrix over the coefficient determinant, a clean JEE Advanced method. The coefficient determinant of rows (2,3),(5,4) is 2·4 - 3·5 = 8 - 15 = -7. To find x, replace the first column with the constants (8,13), giving determinant 8·4 - 3·13 = 32 - 39 = -7. Then x = (-7)/(-7) = 1. Option 2 corresponds to the value of y, not x. Option 13/7 misreads the determinant ratio. Option 8/2 ignores the determinant structure entirely. Hence x = 1. Plausibility check: substituting x = 1 into 2x + 3y = 8 gives 3y = 6 so y = 2, and checking 5(1) + 4(2) = 13 confirms the solution, validating Cramer's rule result for x. Orthogonal matrices preserve lengths, angles and the standard inner product, so geometrically they act only as rotations when the determinant is plus one and as reflections or improper rotations when it is minus one. Their inverse coincides with their transpose, a property that makes them computationally convenient and central to the study of rigid motions and change-of-basis transformations in coordinate geometry.