"Let $A = \{ {a_{ij}}\}$ be a 3 $\times$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i j} \cr } } \right.$$ then $\det (3Adj(2{A^{ - 1}}))$ is equal to _____________."

Question

"Let $A = \{ {a_{ij}}\}$ be a 3 $\times$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr 2 & {if} & {i = j,} \cr {{{( - 1)}^{i + j}}} & {if} & {i > j} \cr } } \right.$$ then $\det (3Adj(2{A^{ - 1}}))$ is equal to _____________."

Accepted Answer

"$$A = \left[ {\matrix{ 2 & { - 1} & 1 \cr { - 1} & 2 & { - 1} \cr 1 & { - 1} & 2 \cr } } \right]$$

$|A| = 4$

$\det (3adj(2{A^{ - 1}}))$

$= {3^3}\left| {adj(2{a^{ - 1}})} \right|$

$= {3^2}{\left| {2{A^{ - 1}}} \right|^2}$

$$ = {3^3}{.2^2}|{A^{ - 1}}{|^2} = {3^3}{.2^2}.{1 \over {|A{|^2}}} = {3^2}{.2^2}.{1 \over {{4^2}}} = 108$$"

Let $A = \{ {a_{ij}}\}$ be a 3 $\times$ 3 matrix,

where $${a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr 2 & {if} & {i = j,} \cr {{{( - 1)}^{i + j}}} & {if} & {i > j} \cr } } \right.$$

then $\det (3Adj(2{A^{ - 1}}))$ is equal to _____________.

Solution

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Let $A = \{ {a_{ij}}\}$ be a 3 $\times$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr 2 & {if} & {i = j,} \cr {{{( - 1)}^{i + j}}} & {if} & {i > j} \cr } } \right.$$ then $\det (3Adj(2{A^{ - 1}}))$ is equal to _____________.

Solution

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More Matrices and Determinants questions

Drill 25 more like these. Every day. Free.

Let $A = \{ {a_{ij}}\}$ be a 3 $\times$ 3 matrix,

where $${a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr 2 & {if} & {i = j,} \cr {{{( - 1)}^{i + j}}} & {if} & {i > j} \cr } } \right.$$

then $\det (3Adj(2{A^{ - 1}}))$ is equal to _____________.