## Quick Reference

Consider a set of *n* linear equations in *n* unknowns *x*_{1}, *x*_{2},…, *x*_{n}, written in matrix form as **Ax**=**b**. When **A** is invertible, the set of equations has a unique solution **x**=**A**^{−1}**b**. Since **A**^{−1}=(1/det **A**) adj **A**, this gives the solutionwhich may be writtenusing the entries of **b** and the cofactors of **A**. This is Cramer's rule. Note that here the numerator is equal to the determinant of the matrix obtained by replacing the *j*-th column of **A** by the column **b**. For example, this gives the solution of

*ax* + *by* = *h*,*cx* + *dy* = *k*,

when *ad*−*bc* ≠ 0, as

*Subjects:*
Mathematics.