Properties of Determinants and Linear Equation System Using Cramer's Rules

In this article we will develop some fundamental characteristics of determinants. The following theorem shows how elementary row operations on the matrix will affect the determinant value.
Theorem 1.

If A is any matrix squared, then det det (A) = det (At).
Example 1



Theorem 2.
If A and B are quadratic matrices of the same size, then det (AB) = det (A) det (B).
Example 2.
Calculate the determinant value of the matrices below,


We get det(A).det (B) = (1) (- 23) = - 23
Conversely, with a direct calculation, det(AB) = - 23, so det(AB) = det (A).det (B).

Application of Determinants in a Linear Equation System Using Cramer's Rules.

Determining the solution of a system of linear equations can also be done using determinant. The following theorem produces a formula for a system consisting of n equations in unknown numbers. This formula is known as the Cramer rule.

Theorem 3.
If AX = B is a system consisting of n equations in n unknown numbers so that det (A) ≠ 0, then the system has a unique solution. This solution is


Where Aj is a matrix that we get by replacing entries in column-j of A. With entries in the matrix.

Example 3.
Use the Cramer rules to complete


Completion:


Therefore,


Cramer's rules give us an easy method of writing system solutions to linear equations n x n with determinants. However, to calculate the solution, we must calculate n + 1 the number of determinants with order n. Even just counting two of the determinants of this order in general will involve more calculations than the completion of the system by using Gauss elimination.

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