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**Unformatted text preview: **Rajiv Kumar Math II System of Linear Equations Rajiv Kumar Math II Determinant and their properties (i) det (A T ) = det (A) (ii) det (C)= -det (A) if C is determinant of matrix where two rows of A are interchanged. (iii) If two rows or two columns of A are identical then det (A)=0 (iv) If D is obtained by multiplying a row or column of A through a real constant c then det (D)= c det (A) Rajiv Kumar Math II If matrix D is the matrix in which ith row of matrix A is replaced with r i + q r j det (D)= det (A) Rajiv Kumar Math II Upper triangular Matrix : if a ij =0 if i>j , then matrix is called upper triangular. Rajiv Kumar Math II Determinant of a upper triangular matrix is product of its diagonal elements. Determinant of product of two matrices is product of their determinants. Similarly one can define lower triangular Matrices. Rajiv Kumar Math II System of equations brief introduction How to add two vectors in R n Dot product of Two vectors in R n Multiplying a vector with a real number. Multiplying two matrices recollect. Rajiv Kumar Math II If we are solving system of linear algebraic equations ,What are nonlinear system of equations , what is general form of a system of linear algebraic Equations ? Rajiv Kumar Math II Solution by elementary row operations : general system of m linear equations, in n unknowns, is written as m b n x mn a x mj a x m a x m a i b n x in a x ij a x i a x i a b n x n a x j a x a x a b n x n a x j a x a x a ..... 1 ..... 2 2 1 1 ..... 1 ..... 2 2 1 1 2 2 ..... 1 2 ..... 2 22 1 21 1 1 ..... 1 1 ..... 2 12 1 11 Rajiv Kumar Math II m i 2 1 n 2 1 b b b b x x x mn m m in i i n n a a a a a a a a a a a a 2 1 2 1 2 22 21 1 12 11 The above system of equation can be return in matrix form as A x =b, Rajiv Kumar Math II We know that the solution of the system of linear equation does not change if we Interchange any two (equations) Multiply any equation by a non-zero scalar Replace a equation by the sum of itself and a scalar multiple of another equation....

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