By Mark Twain

Twain's account of traveling in Europe, "A Tramp Abroad" (1880), glints with the author's clever observations and hugely opinionated reviews on previous international tradition, and showcases his unheard of skill to combine funny sketches, autobiographical tidbits, and ancient anecdotes in a regularly interesting narrative. forged within the kind of a jogging journey via Germany, Switzerland, France and Italy, "A Tramp Abroad" contains between its adventures a voyage by means of raft down the Neckar and an ascent of Mount Blanc through telescope, in addition to the author's makes an attempt to check paintings - a unconditionally imagined task Twain 'authenticated' along with his personal splendidly primitive photos incorporated during this quantity.

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**Example text**

Finally, operation (3) - 6(2) gives Xi + 3x2 + 3^3 £3 0 = 0 Here the third equation tells us nothing and can be ignored. Now observe that we can assign arbitrary values c and d to the unknowns X4 and x2 respectively, and then use back substitution to find x 3 and x\. Hence the most general solution of the linear system is x± = — 2 — c — 3d, x2 — d, X3 = 1 — - , £4 = c. Since c and d can be given arbitrary values, the linear system has infinitely many solutions. What has been learned from these three examples?

The inverse of a square matrix An n x n matrix A is said to be invertible if there is an n x n matrix B such that AB = In = BA. Then B is called an inverse of A. A matrix which is not invertible is sometimes called singular, while an invertible matrix is said to be non-singular. 9 Show that the matrix 1 3 3 9 is not invertible. If f have , ) were an inverse of the matrix, then we should 18 Chapter One: Matrix Algebra 1 3 \ fa 3 9 \c b\ _ (1 d ~ [0 0 1 which leads to a set of linear equations with no solutions, a + 3c b + 3d 3a + 9c 3b + 9d =1 = 0 =0 =1 Indeed the first and third equations clearly contradict each other.

1 xi - x2 + x3 %i + X2 + x3 Xi + 3X2 + £3 + x4 - x4 =2 =3 — 3^4 = 1 To determine if the system has a solution, we apply certain operations to the equations of the system which are designed to eliminate unknowns from as many equations as possible. The important point about these operations is that, although they change the linear system, they do not change its solutions. We begin by subtracting equation 1 from equations 2 and 3 in order to eliminate x\ from the last two equations. These operations can be conveniently denoted by (2) — (1) and (3) — (1) respectively.