By Prabhat Choudhary

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**Sample text**

Consider next q31. 3 =-1 +0+0+ 18= 17. Consider finally q32. 1 = - 4 + 0 + (- 14) + 6 = - 12. We therefore conclude that AB=[ ~ 41 53 2-Ij o -1 7 6 1 4 2 3 0 -2 3 +~ 17 Example. Consider again the matrices A~P -1 4 3 1 5 0 7 -Ij ~ and B= 1 4 2 3 0 -2 3 137 1. -12 56 Matries Note that B is a 4 x 2 matrix and A is a 3 x 4 matrix, so that we do not have a definition for the "product" BA. We leave the proofs of the following results as exercises for the interested reader. Proposition. (Associative Law) Suppose that A is an mn matrix, B is an np matrix and C is an p x r matrix.

Find the leftmost non-zero column of the matrix; 2. Make sure, by applying row operations of type 2, if necessary, that the first (the upper) entry of this column is non-zero. This entry will be called the pivot entry or simply the pivot; 3. , make them 0) all non-zero entries below the pivot by adding (subtracting) an appropriate multiple of the first row from the rows number 2, 3, ... ,m. We apply the main step to a matrix, then we leave the first row alone and apply the main step to rows 2, ...

Then A(B + C) = AB + AC. (b) Suppose that A and Bare m x n matrices and C is an n x p matrix. Then (A + B)C = AC +BC. Proposition. Suppose that A is an m x n matrix, B is an n x p matrix, and that c E JR. Then c(AB) = (cA)B = A(cB). Systems of Linear Equations Note that the system of linear equations can be written in matrix form as Ax = b, where the matrices A, x and b are given. We shall establish the following important result. Proposition. Every system of linear equations of the form,has either no solution, one solution or infinitely many solutions.