InterviewSolution
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InterviewSolution
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Let `A=[{:(1,-2,1),(-2,3,1),(1,1,5):}]`. Verify that ltbtgt (i) `[adjA]^(-1)=adj (A^(-1))` (ii) `(A^(-1)^(-1)=A` |
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Answer» `A=[{:(1,-2,1),(-2,3,1),(1,1,5):}]` `therefore" "|A|=[{:(1,-2,1),(-2,3,1),(1,1,5):}]` `=1(15-1)-(-2)(-10-1)+1(-2-3) ` `=14-22-5-13ne0` `A_(11)=14, A_(12)=11, A_(13)=-5` `A_(21)=11, A_(22)=4, A_(23)=-3` `A_(31)=-5, A_(32)=-3, A_(33)=-1` `therefore" "A=[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]=[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]` `"and A"^(-1)=1/|A|="adj A"=-1/13[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]=[{:(-14/13,-11/13,5/13),(122/12,-4/13,3/13),(5/13,3/13,1/13):}]` (i) `Let B = adj A=[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]rArr|B|=[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]` =14(-4-9)=11(-11-15)-5(-33+20) `=-182+286+65=169ne0` `B_(11)=-13, B_(12)=26, B_(13)=-13` `B_(21)=26, B_(22)=-39, B_(23)=-13` `B_(31)=-13, B_(32)=-13, B_(33)=-65` `therefore" adj B"=[{:(-13,26,-13),(26,-39,-13),(-13,-13,-65):}]=[{:(-13,26,-13),(26,-39,-13),(-13,-13,-65):}]` `"andB"^(-1)=1/|B|"adj B="1/169[{:(-13,26,-13),(26,-39,-13),(-13,-13,-65):}]=1/13[{:(-1,2,-1),(1,-3,-1),(-1,-1,-5):}]` `"Let C"=A^(-1)=[{:(-14/13,-11/13,5/13),(-11/13,-4/13,3/13),(5/13,3/13,1/13):}]` `C_(11)=-13/169=-1/13,C_(12)=26/169=2/13, C_(13)=-13/169=-1/13` `C_(21)=26/169=2/13, C_(22)=39/169=-3/13, C_(23)=-13/169=-1/13` `C_(31)=-13/169=1/13, C_(32)=-13/169=-1/13, C_(33)=-65/169=-5/13` `:."adjC=adjA"^(-1)=[{:(-1/13,2/13,-1/13),(2/13,-3/13,-1/13),(-1/13,-1/13,-5/13):}]=[{:(-1/13,2/13,-1/13),(2/13,-3/13,-1/13),(-1/13,-1/13,-5/13):}]` `=1/13[{:(-1,2,-1),(2,-3,-1),(-1,-1,-5):}]` (ii) `|A^(-1)|=14/13(=4/169-9/169)+11/13(-11/169-15/169)+5/13(-33/169+20/169)=1-1/13` `therefore (A^(-1))^(-1)=1/|A^(-1)|"adj"(A^(-1))` `1/(-1/13)-1/13[{:(-1,2,-1),(2,-3,-1),(-1,-1,-5):}]=[{:(1,-2,1),(2,3,1),(-1,1,5):}]=A` |
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