# Exercises 8,10,12,14, find AB and BA, if possible, or state " underfined". # 8. # > with(linalg); [BlockDiagonal, GramSchmidt, JordanBlock, Wronskian, add, addcol, addrow, adj, adjoint, angle, augment, backsub, band, basis, bezout, blockmatrix, charmat, charpoly, col, coldim, colspace, colspan, companion, concat, cond, copyinto, crossprod, curl, definite, delcols, delrows, det, diag, diverge, dotprod, eigenvals, eigenvects, entermatrix, equal, exponential, extend, ffgausselim, fibonacci, frobenius, gausselim, gaussjord, genmatrix, grad, hadamard, hermite, hessian, hilbert, htranspose, ihermite, indexfunc, innerprod, intbasis, inverse, ismith, iszero, jacobian, jordan, kernel, laplacian, leastsqrs, linsolve, matrix, minor, minpoly, mulcol, mulrow, multiply, norm, normalize, nullspace, orthog, permanent, pivot, potential, randmatrix, randvector, rank, ratform, row, rowdim, rowspace, rowspan, rref, scalarmul, singularvals, smith, stack, submatrix, subvector, sumbasis, swapcol, swaprow, sylvester, toeplitz, trace, transpose, vandermonde, vecpotent, vectdim, vector] > A := matrix(3,3,[-1,0,2,2,1,3,3,0,0]); [ -1 0 2 ] [ ] A := [ 2 1 3 ] [ ] [ 3 0 0 ] -------------------------------------------------------------------------------- > B := matrix(3,3,[4,0,-1,3,2,0,0,2,-4]); [ 4 0 -1 ] [ ] B := [ 3 2 0 ] [ ] [ 0 2 -4 ] -------------------------------------------------------------------------------- > AB := evalm(A &* B); [ -4 4 -7 ] [ ] AB := [ 11 8 -14 ] [ ] [ 12 0 -3 ] -------------------------------------------------------------------------------- > BA := evalm(B &* A); [ -7 0 8 ] [ ] BA := [ 1 2 12 ] [ ] [ -8 2 6 ] -------------------------------------------------------------------------------- # 10. # > A := matrix(1,3,[1,3,-3]); A := [ 1 3 -3 ] -------------------------------------------------------------------------------- > B := matrix(3,1,[0,4,-2]); [ 0 ] [ ] B := [ 4 ] [ ] [ -2 ] -------------------------------------------------------------------------------- > AB := evalm(A &* B); AB := [ 18 ] -------------------------------------------------------------------------------- > BA := evalm(B &* A); [ 0 0 0 ] [ ] BA := [ 4 12 -12 ] [ ] [ -2 -6 6 ] -------------------------------------------------------------------------------- # 12. # > A := matrix(2,2,[5,3,-1,0]); [ 5 3 ] A := [ ] [ -1 0 ] -------------------------------------------------------------------------------- > B := matrix(3,2,[4,-2,-2,0,9,1]); [ 4 -2 ] [ ] B := [ -2 0 ] [ ] [ 9 1 ] -------------------------------------------------------------------------------- > AB := evalm(A &* B); Error, (in linalg[multiply]) matrix dimensions incompatible # AB is underfined. -------------------------------------------------------------------------------- > BA := evalm(B &* A); [ 22 12 ] [ ] BA := [ -10 -6 ] [ ] [ 44 27 ] -------------------------------------------------------------------------------- # 14. # > A := matrix(2,1,[2,1]); [ 2 ] A := [ ] [ 1 ] > B := matrix(2,3,[3,0,-4,0,5,2]); [ 3 0 -4 ] B := [ ] [ 0 5 2 ] -------------------------------------------------------------------------------- > AB := evalm(A &* B); Error, (in linalg[multiply]) matrix dimensions incompatible -------------------------------------------------------------------------------- > BA := evalm(B &* A); Error, (in linalg[multiply]) matrix dimensions incompatible # Both AB and BA are underfined. # # 24. # a). > eqns :={5*x-3*y-2*z=4, 2*x+4*y-z=2,y+7*z=3}; eqns := {5 x - 3 y - 2 z = 4, y + 7 z = 3, 2 x + 4 y - z = 2} -------------------------------------------------------------------------------- > subs(x=3*p-q, y=2*p+4*q, z=-p+2*q,eqns); {- 5 p + 18 q = 3, 11 p - 21 q = 4, 15 p + 12 q = 2} -------------------------------------------------------------------------------- > simplify("); {- 5 p + 18 q = 3, 11 p - 21 q = 4, 15 p + 12 q = 2} -------------------------------------------------------------------------------- > A := matrix(3,2,[11,-21,15,12,-5,18]); [ 11 -21 ] [ ] A := [ 15 12 ] [ ] [ -5 18 ] > X := matrix(2,1,[p,q]); [ p ] X := [ ] [ q ] > B := matrix(3,1,[4,2,3]); [ 4 ] [ ] B := [ 2 ] [ ] [ 3 ] # The associated matrix equation is AX=B. -------------------------------------------------------------------------------- # b). # The associated matrix equations are > A1 := matrix(3,3,[5,-3,-2,2,4,-1,0,1,7]); [ 5 -3 -2 ] [ ] A1 := [ 2 4 -1 ] [ ] [ 0 1 7 ] -------------------------------------------------------------------------------- > X1 := matrix (3,1,[x,y,z]); [ x ] [ ] X1 := [ y ] [ ] [ z ] -------------------------------------------------------------------------------- > B1 := matrix (3,1,[4,2,3]); [ 4 ] [ ] B1 := [ 2 ] [ ] [ 3 ] > eqn := A1 &* X1 = B1; eqn := A1 &* X1 = B1 # The associated matrix equation is A1*X1=B1. -------------------------------------------------------------------------------- > C := matrix(3,2,[3,-1,2,4,-1,2]); [ 3 -1 ] [ ] C := [ 2 4 ] [ ] [ -1 2 ] -------------------------------------------------------------------------------- > T := matrix (2,1,[p,q]); [ p ] T := [ ] [ q ] > eqn1 := X1=C &* T; eqn1 := X1 = C &* T -------------------------------------------------------------------------------- > subs(eqn1,eqn); A1 &* (C &* T) = B1 -------------------------------------------------------------------------------- > coef :=evalm( A1 &* C); [ 11 -21 ] [ ] coef := [ 15 12 ] [ ] [ -5 18 ] # # So, we obtain the coef*X = B as the same result from a). # # 26. # > A := matrix(4,2,[7,5,10,7,14,7,12,7]); [ 7 5 ] [ ] [ 10 7 ] A := [ ] [ 14 7 ] [ ] [ 12 7 ] -------------------------------------------------------------------------------- > B := matrix(2,2,[150,90,180,100]); [ 150 90 ] B := [ ] [ 180 100 ] -------------------------------------------------------------------------------- > C := evalm(A &* B); [ 1950 1130 ] [ ] [ 2760 1600 ] C := [ ] [ 3360 1960 ] [ ] [ 3060 1780 ] -------------------------------------------------------------------------------- # The total dollars sales of Feb. is $1950, Mar. $2760, Apr. $3360, May $3060. # The total costs for Feb. is $1130, Mar. $1600, Apr. $1960, May $1780.