16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-172. Multiplication of a row by a non-zero constant.Let c be a non-zero constant, and let E c,i denote the matrix whose j−throw is the unit vector i j if j ≠ i, and whose i−th row is ce i .3. Addition of a multiple of row i to row j.Let E ci+j denote the matrix obtained from I by adding c times row ito row j.Let us consider some examples.Example 6 Consider an interchange of the first two rows.A =⎡⎢⎣2 3 11 −1 10 4 3⎤⎥⎦ , B =The associated permutation matrix isThen, one sees thatP 12 =⎡⎢⎣⎡⎢⎣0 1 01 0 00 0 1B = P 12 A1 −1 12 3 10 4 3Example 7. Consider the replacing row 1 by row 1 plus 2 times row 3A =⎡⎢⎣2 3 11 −1 10 4 3⎤⎥⎦ , B =The associated matrix is E 23+1 , withandE 23+1 =⎡⎢⎣⎡⎢⎣1 0 20 1 00 0 1B = E 23+1 A.⎤⎥⎦2 11 71 −1 10 4 3⎤⎥⎦ ,⎤⎥⎦⎤⎥⎦