Let for `A=[(1,0,0),(2,1,0),(3,2,1)]`, there be three row matrices `R_(1), R_(2)` and `R_(3)`, satifying the relations, `R_(1)A=[(1,0,0)], R_(2)A=[(2,3,0)]` and `R_(3)A=[(2,3,1)]`. If B is square matrix of order 3 with rows `R_(1), R_(2)` and `R_(3)` in order, then The value of det. `(2A^(100) B^(3)-A^(99) B^(4))` isA. `-27`B. `-9`C. `-3`D. 9

Let for `A=[(1,0,0),(2,1,0),(3,2,1)]`, there be three row matrices `R_

1.	Let for `A=[(1,0,0),(2,1,0),(3,2,1)]`, there be three row matrices `R_(1), R_(2)` and `R_(3)`, satifying the relations, `R_(1)A=[(1,0,0)], R_(2)A=[(2,3,0)]` and `R_(3)A=[(2,3,1)]`. If B is square matrix of order 3 with rows `R_(1), R_(2)` and `R_(3)` in order, then The value of det. `(2A^(100) B^(3)-A^(99) B^(4))` isA. `-27`B. `-9`C. `-3`D. 9
Answer» Correct Answer - A `overset(B)([(-,R_(1),-),(-,R_(2),-),(-,R_(3),-)])overset(A)([(1,0,0),(2,1,0),(3,2,1)])=overset(C)([(1,0,0),(2,3,0),(2,3,1)])` (1) `:.` (det. B) (det. A)=3 `:.` (det. B)=3 [as det. A=1] det. `(2A^(100)B^(3)-A^(99)B^(4))` = det. `(A^(99) (2A-B)B^(3))` `=("det. A")^(99)xxdet. (2A-B)xx("det B")^(3)` Now from (1), we get `B=A^(-1) C=[(1,0,0),(-2,1,0),(1,-2,1)][(1,0,0),(2,3,0),(2,3,1)]` `=[(1,0,0),(0,3,0),(-1,-3,1)]` `:. 2A-B=[(1,0,0),(4,-1,0),(7,7,1)]` `:.` det. `(2A-B)=-1` `:.` det. `(2A^(100) B^(3)-A^(99) B^(4))=(1)^(99) (-1) (3)^(3)=-27`

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