Solving the linear equations with the help of the inverse matrix and Cramer’s methods

Solving the linear equations with the help of the inverse matrix and Cramer’s methods

Prepared by: Mokhinur Raupova

CSPU
MI-23/6

Cramer’s method

Cramer’s method

• Suppose a system of n linear equations in the n variables x1, x2, . . . , xn is equivalent to the matrix equation DX = B, and |D| ≠ 0.
• Then, its solutions are: where Dxi is the matrix obtained by replacing the ith column of D by the n x 1 matrix B.

Use Cramer’s Rule to solve the system

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution
Example:
Process & Solution:

• First, we evaluate the determinants that appear in Cramer’s Rule

Cramer’s method

Note that D is the coefficient matrix and that Dx, Dy, and Dz are obtained by replacing the first, second, and third columns of D by the constant terms.

Solution

Now, we use Cramer’s Rule to get the solution:

Let’s have a practise!

Inverse Matrices and Systems of Equations
Example:
Write the system as a matrix equation
Coefficient matrix
Constant matrix
Variable matrix

Inverse Matrices and Systems of Equations

Step 1: Write a matrix equation and find its determinant
Det(A) = 2·2 – 3·1 = 1 ≠ 0

Inverse Matrices and Systems of Equations

Step 2: Find the inverse matrix of A
Step 3: Solve for the variable matrix
The solution to the system is (4, 1).
Example:
A
B
X

Let’s have a practise!

Assignment

Thank you for your attention 