Solving systems of equations using matrices

Matrices and determinants

A system of linear equations is a set of n equations in n unknowns (variables) of the form

where, the unknowns are denoted by x₁, x₂, . . . x_n, the coefficients (a_ij) and constants (b_i) are given values. The system of equations above can be written in matrix form as

Ax = b

where, A is matrix of the coefficients (a_ij), x is a vector of n unknowns and b is a vector of n constants.

Inverse matrix method

The matrix equation Ax = b, where A is an n ´ n regular matrix (det(A) is not 0),

multiplied by A^-1 gives A^-1Ax = A^-1b

since, A^-1A = I by definition and Ix = x, where I is the identity matrix, as

then, x = A^-1b.

Thus, the inverse matrix method uses this matrix equation to find the solution to the system of equations directly.

Example: Find the solution of the given three equations using the inverse matrix method.

Solution:

Given equations are written as the equivalent matrix equation. Then, both sides of the above matrix equation we multiply by the inverse of the coefficient matrix A (calculation of which is shown in the previous example),

Since. A^-1A = I and Ix = x, then

Therefore, the solution is, x₁ = 1, x₂ = 2 and x₃ = 3.

Cramer’s rule (using the determinant) to solve systems of linear equations

Solving system of two equations in two unknowns using Cramer's rule

A system of two equations in two unknowns, the solution to a system by Cramer’s rule (use of determinants).

the solution to the system

Solving system of three equations in three unknowns using Cramer's rule

A determinant of rank n can be evaluated by expanding to its cofactors of rank n - 1, along any row or column taking into account the scheme of the signs.

For example, the determinant of rank n = 3,

Example: Solve the system of three equations in three unknowns using method of expanding to cofactors.

Solution:

Method of expanding a determinant of a rank n by cofactors, example

The value of a determinant will not change by adding multiples of any column or row to any other column or row. This way created are zero entries that simplify subsequent calculations.

Example: An application of the method of expanding a determinant to cofactors to evaluate the determinant of the rank four.

Added is third to the second colon. Then, the second row multiplied by -3 is added to the first row. The obtained determinant is then expanded to its cofactors along the second colon,

The first colon multiplied by -1 is added to the third colon. The obtained determinant is then expanded along the third colon.

Cramer’s Rule and inverse matrix method correlation

Cramer's Rule says that a system of n linear equations in n unknowns,

will have a unique solution if the determinant of the coefficient matrix det(A) = A = |a_ik| is nonzero, and

in this case, the value of the unknown x_k is given by the expression

where the numerator A_k is equal to the determinant A of the matrix A that results when the k^th column (the coefficients of the unknown to be found) of the coefficient matrix is replaced by the column of constants, b₁, b₂, . . . , b_n.

Let's show that Cramer's Rule, to find the value of the unknown x_k, coincide with the solution given by the matrix equation x = A^-1b.

That is, by expanding the determinant A_k by the k^th column we get

A_k = b₁A₁_k + b₂A₂_k + . . . + b_nA_n_k

where, A₁_k, A₂_k, . . . , A_n_k are the cofactors of the entries, b₁, b₂, . . . , b_n, that are the same as the cofactors of the entries, a₁_k, a₂_k, . . . , a_n_k, of the determinant A = det(A) or the matrix A.

To find the value of unknown x_k from x = A^-1b we should calculate the scalar product of the k^th row vector of A^-1 and the column vector b.

Therefore, if a_i_k denotes the entries of A^-1 then

x_k = a_k₁b₁ + a_k₂b₂ + . . . + a_k_nb_n.

Recall that

where A_ki denotes the ki^th entry of the transpose of the cofactor matrix.

As the transposition interchange rows and columns, the above expression for the unknown x_k can also be written as

what coincide with

where A_k = b₁A₁_k + b₂A₂_k + . . . + b_nA_n_k.

That proves the correlation between solution given by Cramer's Rule and the solution given by the matrix equation x = A^-1b.

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