Cramer's Rule

Cramer's Rule

Cramer's Rule is a method used to solve systems of linear equations. Specifically, it provides a formula for the solutions of a system of n linear equations in n unknowns, provided that the coefficient matrix of the system is invertible.

Suppose we have a system of n linear equations in n unknowns, given by:

a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1

a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_2

...

a_{n1}x_1 + a_{n2}x_2 + ... + a_{nn}x_n = b_n

where a_{ij} are the coefficients of the variables, x_i are the unknowns, and b_i are the constant terms.

We can represent this system using matrices as follows:

AX = B

where A is the coefficient matrix, X is the column matrix of unknowns, and B is the column matrix of constants. If A is invertible, then we can solve for X as follows:

X = A^{-1}B

Cramer's Rule provides a formula for the solutions of this system in terms of the determinants of certain submatrices of A.

Suppose we define the matrix A_i as the matrix obtained by replacing the i-th column of A with the column matrix B. Then, the i-th component of X can be expressed as:

x_i = det(A_i) / det(A)

where det(A_i) is the determinant of the matrix A_i and det(A) is the determinant of the coefficient matrix A.

To see why this is true, note that multiplying A by X gives us:

a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1

a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_2

...

a_{n1}x_1 + a_{n2}x_2 + ... + a_{nn}x_n = b_n

which can be written as:

AX = B

If we replace the i-th column of A with the column matrix B, we obtain:

A_i X_i = B_i

where X_i is the column matrix obtained by replacing the i-th component of X with b_i and B_i is the column matrix obtained by replacing the i-th column of B with the column matrix X_i.

The determinant of A_i can be expressed as a linear combination of the determinants of A and certain submatrices of A, as follows:

det(A_i) = det(a_1,1, a_1,2, ..., a_1,i-1, b_1, a_1,i+1, ..., a_1,n)

- det(a_2,1, a_2,2, ..., a_2,i-1, b_2, a_2,i+1, ..., a_2,n)

+ det(a_3,1, a_3,2, ..., a_3,i-1, b_3, a_3,i+1, ..., a_3,n)

- ...

+ (-1)^{i+n} det(a_n,1, a_n,2, ..., a_n,i-1, b_n, a_n,i+1, ..., a_n,n)

where the signs alternate and the submatrices exclude