If A is a square matrix of order 3 such that det ( A ) = 3 and det ( adj ( - 4 ? adj ( - 3 ? adj ( 3 ? adj ( ( 2 A ) - 1 ) ) ) ) ) = 2 m 3 n . Then m + 2 n is equal to:

Question

If A is a square matrix of order 3 such that $\det (A) = 3$ and $\det (adj (- 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1}))))) = 2^{m} 3^{n} .$ Then $m + 2 n$ is equal to:

Smart Achievers · Accepted Answer

(4)

$| A |$ $= 3$

$| adj (- 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1})))) |$

$=$ $| - 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1}))) |^{2}$

$= 4^{6}$ $| adj (- 3 adj (3 adj ({(2 A)}^{- 1}))) |^{2}$

$= 2^{12} \cdot 3^{12}$ $| 3 adj {(2 A)}^{- 1} |^{8}$

$= 2^{12} \cdot 3^{12} \cdot 3^{24}$ $| adj {(2 A)}^{- 1} |^{8}$

$= 2^{12} \cdot 3^{36}$ $| {(2 A)}^{- 1} |^{16}$

$= 2^{12} \cdot 3^{36} \frac{1}{| 2 A |^{16}}$

$= 2^{12} \cdot 3^{36} \frac{1}{2^{48} | A |^{16}}$

$= 2^{12} \cdot 3^{36} \frac{1}{2^{48} 3^{16}}$

$= \frac{3^{20}}{2^{36}} = 2^{- 36} \cdot 3^{20}$

$m = - 36,$ $n = 20$

$m + 2 n = 4$

Solution

Q.
If A is a square matrix of order 3 such that $\det (A) = 3$ and $\det (adj (- 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1}))))) = 2^{m} 3^{n} .$ Then $m + 2 n$ is equal to:

1 2

2 3

3 6

4 4

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Solution

Q. If A is a square matrix of order 3 such that det(A)=3 and det(adj(-4 adj(-3 adj(3 adj((2A)-1)))))=2m3n. Then m+2n is equal to:

1 2

2 3

3 6

4 4

Ans. (4) |A|=3 |adj(-4adj(-3adj(3adj((2A)-1))))| =|-4adj(-3adj(3adj((2A)-1)))|2 =46|adj(-3adj(3adj((2A)-1)))|2 =212·312|3adj(2A)-1|8 =212·312·324|adj(2A)-1|8 =212·336|(2A)-1|16 =212·3361|2A|16 =212·3361248|A|16 =212·3361248316 =320236=2-36·320 m=-36, n=20 m+2n=4

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Q.
If A is a square matrix of order 3 such that $\det (A) = 3$ and $\det (adj (- 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1}))))) = 2^{m} 3^{n} .$ Then $m + 2 n$ is equal to: