Question

a1 a12 a13 a14 bi by b 2 Denote row i in matrix A above as vector a' and row i in matrix B as vector bn' for example, a aan a3 aul Similarly, denote column k in matrix A as vector and column k in matrix B as vector b. a) Does matrix C AB exist? If no, explain why not. If yes, write it out expressing each element ck as the inner product of the relevant vectors defined above b) Does matrix D- BA exist? If no, explain why not. If yes, write it out expressing each element dak as the inner product of the relevant vectors defined above 2) Find the length of vector i -2 s 3) Let u-33 2], and w-Is 1 a) Treating u, v, and w, as vectors, are the inner products us", v.u, and u.w, defined? If yes, compute them. If any of them is not defined, explain why not. b) Treating u, v', and w, as matrices, are the products in,,, v,u, and u,v, defined? If ye compute them. If any of them is not defined, explain why not.

Answer 1

A = [a_11 a_12 a_13 a_14 a_21 a_22 a_23 a_24 a_31 a_32 a_33 a_34] B = [b_11 b_12 b_13 b_21 b_22 b_23 b_31 b_32 b_33 b_41 b_42 b_43] C = AB order of matrix A = 3 times 4 order of matrix B = 4 times 3 To find AB No. of columns of A = No of rows of B 4 = 4 so, AB Exists Elements of AB are AB = [a_11 a_12 a_13 a_14 a_21 a_22 a_23 a_24 a_31 a_32 a_33 a_34] [b_11 b_12 b_13 b_21 b_22 b_23 b_31 b_32 b_33 b_41 b_32 b_33 b_41 b_42 b_43] = a_11 b_11 + a_12 b_21 + a_13 b_31 + a_14 b_41 a_11 b_12 + a_12 b_22 + a_13 b_32 + a_14 b_42 a_21 b_11 + a_22 b_21 + a_23 b_31 + a_24 b_41 a_21 b_12 + a_22 b_22 + a_23 b_32 + a_24 b_42 a_31 b_11 + a_32 b_21 + a_33 b_31 + a_34 b_41 a_31 b_12 + a_32