Understanding Linear Equations and Matrix Operations
Explore the concepts of linear equations, matrix forms, determinants, and finding solutions for variables like x1, x2, x3. Learn about Cramer's Rules, Adjoint Matrix, and calculating the inverse of a matrix through examples and formulas.
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Example : Find x1and x2from these equation : Solution :
General form : Where : a1, a2, a3.. anand b are constantas x1, x2, x3.. xnare variables
Linear Equation in matrix form If we have some equations : Then, we can write :
General form : Where : Or, we can write :
Example : Find x1and x2from these equation : Solution : Find A-1....
A-1 : Formula :
Cramers rules Assume : Determinants :
x1 : x2 : x3 :
Example : Find x1 and x2 from these equation : Solution :
x1 and x2 : Proof :
Adjoint Matrix Adjoint matrix of a square matrix is the transpose of the matrix formed by cofactors of elements of determinant |A| How to calculate adjoint : Calculate minor matrix for each element of matrix Make cofactor matrix cofactor is a sign minor, denoted by : Cij = (-1)ij . Mij Change to Transpose matrix.
Example Find inverse for A : Calculate |A| : =(1.5.3 + 2.0.2 + 3.0.4) (3.5.2 +1.0.4 + 2.0.3) = 15 30 = - 15
Make a new matrix with minor and cofactor Transpose that matrix :
Matrix form : Formula :
Determinants : Use minor cofactor :
New matrix K : A-1 :
Questions Find x1 and x2 from these equations : 2x1+ x2 4 = 0 x1 3x2+ 5 = 0 Find x, y and z from these equations : x + y + z 6 = 0 2x z + 1 = 0 x y + 2z 5 = 0
