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A Practical Approach to Linear Algebra by Prabhat Choudhary

By Prabhat Choudhary

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For part (d), we can consider the matrix A' obtained from A by multiplying each entry of A by -1. The theory of multiplication is rather more complicated, and includes multiplication of a matrix by a scalar as well as mUltiplication of two matrices. We first study the simpler case of multiplication by scalars. 52 Matrics Definition. Suppose that the matrix A -- [ a~1 .. ami has m rows and n columns, and that C E JR. Then we write cA = [C~l1 C~ln 1 cam I camn . and call this the product of the matrix A by the scalar c.

It now follows that for every c E lR, we have A(u + c(u - v» = Au + A(c(u - v» = Au + c(A(u - v» = b + c) = b; so that x = u + c(u - v) is a solution for every c E lR. Clearly we have infinitely many solutions. Inversion of Matrices We shall deal with square matrices, those where the number ofrows equals the number of columns. Definition.

Then (a) c(A + B) = cA + cB; (b) (c + d)A = cA + dA; (c) OA = 0; and (d) c(dA) = (cd)A. Proof These are all easy consequences of ordinary multiplication, as multiplication by scalar c is simply entry-wise multiplication by the number c. The question of multiplication of two matrices is rather more complicated. To motivate this, let us consider the representation of a system of linear equations al1 xI +···+a1nxn =~, I .. in the form Ax = b, where a~ A=: [ ami a~n: 1andb = [~I: 1 amn represent the coeffcients and bm 53 Matrics represents the variables.

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