5. (15 points). This is a question about row reduced form. In each part, either
ID: 3138570 • Letter: 5
Question
5. (15 points). This is a question about row reduced form. In each part, either give an example of a 3 x 4 matrix R in row reduced form with the required property, or give a short explanation why no example exists. (a) For all b, the matrix equation Rx-b has at least one solution (b) There exists b such that the matrix equation Rx b has exactly one solution. c)For all matrices A whose row reduced form is R, the columns of A are linearly independent (d) For all matrices A whose row reduced form is R, every column of A is a linear combination of the first and third columns of A.Explanation / Answer
5. a). Let R =
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and b=(2,3,4)T. Also, let X=(x,y,z,w)T. Then Rx =b is equivalent to x+w=2 or, x=2-w, y+2w=3 or, y = 3-2w , z+3w = 4 or, z = 4-3w so that X = (2-w,3-2w, 4-3w,w)T = (2,3,4,0)T+w(-1,-2,-3,1)T. Thus, the equation EX = b has infinite solutions.
b). It is not possible for the equation RX = b to have a unique solution since this equation is equivalent to a system of 3 non-homogeneous equations in 4 variables. Thus, either the system will be inconsistent or there will be at least 1 free variable so that there will be infinite solutions.
c). It is not possible for any 3x4 matrix A to have linearly independent columns as col(A) is a subspace of R3 which has dimension 3 so that more than 3 vectors in R3 will be necessarily linearly dependent.
d). Let A =
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The 2nd column of A is c2 = c1 +c3 and the 4th column of A is c4= 2c1+c3 where c1,c2,c3,c4 are the 1st,2nd,3rd and the 4th columns of A.
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