InterviewSolution
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InterviewSolution
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Obtain the inverse of the matrix `A=[{:(2,3),(1,1):}]` using elementary operations. |
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Answer» Write `A=IA. i.e. [{:(,0,1,2),(,1,2,3),(,3,1,1):}]=[{:(,1,0,0),(,0,1,0),(,0,1,1):}]A` or `[{:(,1,2,3),(,0,1,2),(,3,1,1):}]=[{:(,0,1,0),(,1,0,0),(,0,0,1):}]A ("applying "R_(1) Leftrightarrow R_(2)` `or [{:(,1,2,3),(,0,1,2),(,0,-5,-8):}]=[{:(,0,1,0),(,1,0,0),(,0,-3,1):}]A ("applying "R_(3) rightarrow R_(3)-3R_(1))` or `or [{:(,1,0,-1),(,0,1,2),(,0,-5,-8):}]=[{:(,-2,1,0),(,1,0,0),(,0,-3,1):}]A ("applying "R_(1) rightarrow R_(1)-2R_(2))` `or [{:(,1,0,-1),(,0,1,2),(,0,0,2):}]=[{:(,-2,1,0),(,1,0,0),(,5,-3,1):}]A ("applying "R_(3) rightarrow R_(3)+5R_(2))` `or [{:(,1,0,-1),(,0,1,2),(,0,0,1):}]=A[{:(,-2,1,0),(,1,0,0),(,(5)/(2),(-3)/(2),(1)/(2)):}]A ("applying "R_(3) rightarrow (1)/(2)R_(3))` `or [{:(,1,0,0),(,0,1,2),(,0,0,1):}]=A[{:(,(1)/(2),(-1)/(2),(1)/(2)),(,1,0,0),(,(5)/(2),(-3)/(2),(1)/(2)):}]A ("applying "R_(1) rightarrow R_(1)+R_(3))` `or [{:(,1,0,0),(,0,1,0),(,0,0,1):}]=A[{:(,(1)/(2),(-1)/(2),(1)/(2)),(,-4,3,-1),(,(5)/(2),(-3)/(2),(1)/(2)):}]A ("applying "R_(2) rightarrow R_(2)-2R_(3))` `"Hence"=A^(-1)[{:(,(1)/(2),(-1)/(2),(1)/(2)),(,-4,3,-1),(,(5)/(2),(-3)/(2),(1)/(2)):}]` |
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