6
Question
Show that
(i) [5−167][213 4]≠[2134][5−167]
(ii) ⎡⎢⎣100211300⎤⎥⎦⎡⎢⎣−1021−13014⎤⎥⎦≠⎡⎢⎣−1021−13014⎤⎥⎦⎡⎢⎣100211300⎤⎥⎦
(i) [5−167][213 4]≠[2134][5−167]
(ii) ⎡⎢⎣100211300⎤⎥⎦⎡⎢⎣−1021−13014⎤⎥⎦≠⎡⎢⎣−1021−13014⎤⎥⎦⎡⎢⎣100211300⎤⎥⎦
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Solution
(i) [5−167][2134]
=[5(2)−1(3)5(1)−1(4)6(2)+7(3)6(1)+7(4)]
=[10−35−412+216+28]=[713334]
[2134][5−167]
=[2(5)+1(6)2(−1)+1(7)3(5)+4(6)3(−1)+4(7)]
=[10+6−2+715+24−3+28]=[1653925]
∴[5−167][2134]≠[2134][5−167]
(ii) ⎡⎢⎣100211300⎤⎥⎦⎡⎢⎣−1021−13014⎤⎥⎦
=⎡⎢⎣1(−1)+2(0)+3(2) 0(−1)+1(0)+0(2)1(−1)+1(0)+0(2) 1(1)+2(−1)+3(3)0(1)+1(−1)+0(3)1(1)+1(−1)+0(3) 1(0)+2(1)+3(4)0(0)+1(1)+0(4)1(0)+1(1)+0(4)⎤⎥⎦
=⎡⎢⎣50−18−101411⎤⎥⎦
⎡⎢⎣−1021−13014⎤⎥⎦⎡⎢⎣100211300⎤⎥⎦
=⎡⎢⎣1(1)+1(0)+0(1)0(1)+(−1)(0)+1(1)2(1)+3(0)+4(1)−1(2)+1(1)+0(1)0(2)+(−1)(1)+1(1)2(2)+3(1)+4(1)−1(3)+1(0)+0(0)0(3)+(−1)(0)+1(0)2(3)+3(0)+4(0)⎤⎥⎦
=⎡⎢⎣−116−1011−306⎤⎥⎦
∴⎡⎢⎣100211300⎤⎥⎦⎡⎢⎣−1021−13014⎤⎥⎦≠⎡⎢⎣−1021−13014⎤⎥⎦⎡⎢⎣100211300⎤⎥⎦
=[5(2)−1(3)5(1)−1(4)6(2)+7(3)6(1)+7(4)]
=[10−35−412+216+28]=[713334]
[2134][5−167]
=[2(5)+1(6)2(−1)+1(7)3(5)+4(6)3(−1)+4(7)]
=[10+6−2+715+24−3+28]=[1653925]
∴[5−167][2134]≠[2134][5−167]
(ii) ⎡⎢⎣100211300⎤⎥⎦⎡⎢⎣−1021−13014⎤⎥⎦
(ii) ⎡⎢⎣100211300⎤⎥⎦⎡⎢⎣−1021−13014⎤⎥⎦
=⎡⎢⎣1(−1)+2(0)+3(2) 0(−1)+1(0)+0(2)1(−1)+1(0)+0(2) 1(1)+2(−1)+3(3)0(1)+1(−1)+0(3)1(1)+1(−1)+0(3) 1(0)+2(1)+3(4)0(0)+1(1)+0(4)1(0)+1(1)+0(4)⎤⎥⎦
=⎡⎢⎣50−18−101411⎤⎥⎦
⎡⎢⎣−1021−13014⎤⎥⎦⎡⎢⎣100211300⎤⎥⎦
=⎡⎢⎣1(1)+1(0)+0(1)0(1)+(−1)(0)+1(1)2(1)+3(0)+4(1)−1(2)+1(1)+0(1)0(2)+(−1)(1)+1(1)2(2)+3(1)+4(1)−1(3)+1(0)+0(0)0(3)+(−1)(0)+1(0)2(3)+3(0)+4(0)⎤⎥⎦
=⎡⎢⎣−116−1011−306⎤⎥⎦
∴⎡⎢⎣100211300⎤⎥⎦⎡⎢⎣−1021−13014⎤⎥⎦≠⎡⎢⎣−1021−13014⎤⎥⎦⎡⎢⎣100211300⎤⎥⎦
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