Question

la 2 2) 11. If the matrix A=121 b 2 cl obeys the law AA = I, find a, b and c.(Here A' is the transpose of A and I is the unit matrix of order 3).

JEE/Engineering Exams

Maths

Solution

161

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( A=frac{1}{3}left[begin{array}{ccc}a & 2 & 2 2 & 1 & b 2 & c & 1end{array}right] quad begin{array}{cc}frac{d}{9}left(1+c^{2}+1right)=1end{array} ) ( = ) ( =4 ) ( =+2 ) ( A=frac{1}{3}left[begin{array}{lll}a & 2 & 2 2 & 1 & c 2 & b & 1end{array}right] quad c=pm 2 ) ( = ) ( frac{1}{3} times frac{1}{3}left[begin{array}{ccc}a & 2 & 2 2 & 1 & b 2 & c & 1end{array}right]left[begin{array}{ccc}9 & 2 & 2 2 & 1 & c 2 & b & 1end{array}right]=left[begin{array}{ccc}1 & 0 & 0 0 & 1 & 0 0 & 0 & 1end{array}right] ) 5 ( frac{1}{9}left[begin{array}{lll}a^{2}+4+4 & 2 a+2+2 b & 2 a+2 c+2 2 a+2+2 b & 4+1+b^{2} & 4+c+b 2 a+2 c+2 & 4+c+b & 4+c^{2}+1end{array}right]=left[begin{array}{lll}1 & 0 & 0 0 & 1 & 0 0 & 0 & 1end{array}right] ) Comparing the cossesponding tecms, begin{tabular}{c|c} ( frac{1}{9}left(a^{2}+4+4right)=1 ) & ( frac{1}{9}left(4+1+b^{2}right)=1 ) 3 & ( a^{2}+8 ) & ( b^{2}+5=9 ) 3 & ( a^{2}=1 ) & ( b^{2}=4 ) ( a ) & ( =pm 1 ) & ( b=pm 2 ) end{tabular}