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
