his question come from Bradley (2008, pp. 495–496, Progress Ex-ercises 9.2). Let I2 be the (2 × 2) identity matrix, and consider thefollowing three matrices:A =( 1 −40 9), B =( 4 3−7 0), and C =( 5 −1 −112 0 2).(a) If possible, find A + B.(b) If possible, find A − B.(c) If possible, find A + 4B.(d) If possible, find A + I2.(e) If possible, find AI2.(f) If possible, find A + C.(g) If possible, find A + BT .(h) If possible, find BC.(i) If possible, find CB.(j) If possible, find CBT .(k) If possible, find (AB)T .(l) If possible, find C + 5I2.(m) If possible, find CT A.(n) If possible, find (BC)T .(o) If possible, find AC + B
Question
his question come from Bradley (2008, pp. 495–496, Progress Ex-ercises 9.2). Let I2 be the (2 × 2) identity matrix, and consider thefollowing three matrices:A =( 1 −40 9), B =( 4 3−7 0), and C =( 5 −1 −112 0 2).(a) If possible, find A + B.(b) If possible, find A − B.(c) If possible, find A + 4B.(d) If possible, find A + I2.(e) If possible, find AI2.(f) If possible, find A + C.(g) If possible, find A + BT .(h) If possible, find BC.(i) If possible, find CB.(j) If possible, find CBT .(k) If possible, find (AB)T .(l) If possible, find C + 5I2.(m) If possible, find CT A.(n) If possible, find (BC)T .(o) If possible, find AC + B
Solution
Sure, let's solve these step by step:
(a) A + B = [(1+4) (-4+3) (0+(-7)) (9+0)] = [5 -1 -7 9]
(b) A - B = [(1-4) (-4-3) (0-(-7)) (9-0)] = [-3 -7 7 9]
(c) A + 4B = [1 -4 0 9] + 4*[4 3 -7 0] = [1 -4 0 9] + [16 12 -28 0] = [17 8 -28 9]
(d) A + I2 = [1 -4 0 9] + [1 0 0 1] = [2 -4 0 10]
(e) AI2 = A * I2 = [1 -4 0 9] * [1 0 0 1] = [1 -4 0 9]
(f) A + C = [1 -4 0 9] + [5 -1 -112 0 2] = [6 -5 -112 9 2]
(g) A + BT is not possible because the dimensions of A and BT do not match.
(h) BC = [4 3 -7 0] * [5 -1 -112 0 2] = [20 -4 -336 0 6]
(i) CB = [5 -1 -112 0 2] * [4 3 -7 0] = [20 -5 -448 0 2]
(j) CBT is not possible because the dimensions of C and BT do not match.
(k) (AB)T is not possible because the dimensions of A and B do not match for multiplication.
(l) C + 5I2 = [5 -1 -112 0 2] + 5*[1 0 0 1] = [5 -1 -112 0 2] + [5 0 0 5] = [10 -1 -112 0 7]
(m) CT A is not possible because the dimensions of CT and A do not match for multiplication.
(n) (BC)T is not possible because the dimensions of B and C do not match for multiplication.
(o) AC + B is not possible because the dimensions of A and C do not match for multiplication.
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