If A is matrix of order 3 such that | A | = 5 and B = adj A , then the value of | | A - 1 | ( A B ) T | is equal to (where | A | denotes determinant of matrix A . A T denotes transpose of matrix A , A - 1 denotes inverse of matrix A , adj A

Question

If A is matrix of order 3 such that $| A |$ $= 5$ and $B = adj A$ , then the value of $| | A^{- 1} | {(A B)}^{T} |$ is equal to (where $| A |$ denotes determinant of matrix $A . A^{T}$ denotes transpose of matrix $A,$ $A^{- 1}$ denotes inverse of matrix $A$ , $adj A$ denotes adjoint of matrix A)

Smart Achievers · Accepted Answer

(2)

$| A^{- 1} |$ $= \frac{1}{| A |} = \frac{1}{5}$

$| {(A B)}^{T} |$ $=$ $| A B |$ $=$ $| A (adj A) |$ $=$ $A$ $\cdot$ $| adj A |$ $= 5 \times 5^{2} = 5^{3}$

$∴$ $| | A^{- 1} | {(A B)}^{T} |$ $=$ $| \frac{1}{5} {(A B)}^{T} |$ $= \frac{1}{5^{3}}$ $| A B |$ $= 1$

Solution

Q.
If A is matrix of order 3 such that $| A |$ $= 5$ and $B = adj A$ , then the value of $| | A^{- 1} | {(A B)}^{T} |$ is equal to (where $| A |$ denotes determinant of matrix $A . A^{T}$ denotes transpose of matrix $A,$ $A^{- 1}$ denotes inverse of matrix $A$ , $adj A$ denotes adjoint of matrix A)

1 $5$

2 $1$

3 $25$

4 $\frac{1}{25}$

Ans.
(2)

$| A^{- 1} |$ $= \frac{1}{| A |} = \frac{1}{5}$

$| {(A B)}^{T} |$ $=$ $| A B |$ $=$ $| A (adj A) |$ $=$ $| A |$ $\cdot$ $| adj A |$ $= 5 \times 5^{2} = 5^{3}$

$∴$ $| | A^{- 1} | {(A B)}^{T} |$ $=$ $| \frac{1}{5} {(A B)}^{T} |$ $= \frac{1}{5^{3}}$ $| A B |$ $= 1$

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Solution

Q. If A is matrix of order 3 such that |A|=5 and B=adjA, then the value of ||A-1|(AB)T| is equal to (where |A| denotes determinant of matrix A.AT denotes transpose of matrix A, A-1 denotes inverse of matrix A, adjA denotes adjoint of matrix A)

1 5

2 1

3 25

4 125

Ans. (2) |A-1|=1|A|=15 |(AB)T|=|AB|=|A(adjA)|=|A|·|adjA|=5×52=53 ∴ ||A-1|(AB)T|=|15(AB)T|=153|AB|=1