# If ${\text{A}}\left( {{\text{adjA}}} \right) = 5{\text{I}}$ where ${\text{I}}$ is the identity matrix of order 3, then $|adjA|$ is equal to

A. 125

B. 25

C. 5

D. 10

Last updated date: 26th Mar 2023

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Answer

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Hint: Use property of inverse of A and determinant of adjoint of A. Also two matrices are equal to each other then, the order of both the matrices will be equal.

Given, ${\text{A}}\left( {{\text{adjA}}} \right) = 5{\text{I}}$ where order of identity matrix is 3.

Clearly, the order of matrix A and that of identity matrix are equal.

So, the order of matrix A is also 3.

As we know that inverse of any matrix A is given by ${{\text{A}}^{ - 1}} = \dfrac{1}{{|A|}}\left( {{\text{adjA}}} \right)$ where |A| is the determinant of matrix A and adjA is the adjoint matrix of matrix A.

$\therefore {\text{ A}}\left[ {{{\text{A}}^{ - 1}}} \right] = {\text{A}}\left[ {\dfrac{1}{{|A|}}\left( {{\text{adjA}}} \right)} \right] = \dfrac{{{\text{A}}\left( {{\text{adjA}}} \right)}}{{|A|}} = \dfrac{{5{\text{I}}}}{{|A|}}$

Also, we know that ${\text{ A}}\left[ {{{\text{A}}^{ - 1}}} \right] = {\text{I}}$ where ${\text{I}}$ is the identity matrix order 3

Therefore, $

\Rightarrow {\text{I}} = \dfrac{{5{\text{I}}}}{{|A|}} \\

\Rightarrow |{\text{A}}|I = 5I \\

$

On comparing the above equation, we get

Determinant of the matrix A, $|{\text{A}}| = 5$

Using the identity, \[|{\text{adjA}}| = {\left[ {|A|} \right]^{n - 1}}\] where n is the order of the matrix of A

Put $|{\text{A}}| = 5$ and ${\text{n}} = 3$ in the above identity, we have

\[ \Rightarrow |{\text{adjA}}| = {\left[ 5 \right]^{3 - 1}} = {5^2} = 25\]

Therefore, the determinant of matrix adjA is 25.

Option B is correct.

Note- Here, the inverse matrix only exists for non-singular matrices (i.e., determinant of that matrix whose inverse is required should always be non-zero). Also if in an equation two matrices are equal to each other then, order of both the matrices will be equal.

Given, ${\text{A}}\left( {{\text{adjA}}} \right) = 5{\text{I}}$ where order of identity matrix is 3.

Clearly, the order of matrix A and that of identity matrix are equal.

So, the order of matrix A is also 3.

As we know that inverse of any matrix A is given by ${{\text{A}}^{ - 1}} = \dfrac{1}{{|A|}}\left( {{\text{adjA}}} \right)$ where |A| is the determinant of matrix A and adjA is the adjoint matrix of matrix A.

$\therefore {\text{ A}}\left[ {{{\text{A}}^{ - 1}}} \right] = {\text{A}}\left[ {\dfrac{1}{{|A|}}\left( {{\text{adjA}}} \right)} \right] = \dfrac{{{\text{A}}\left( {{\text{adjA}}} \right)}}{{|A|}} = \dfrac{{5{\text{I}}}}{{|A|}}$

Also, we know that ${\text{ A}}\left[ {{{\text{A}}^{ - 1}}} \right] = {\text{I}}$ where ${\text{I}}$ is the identity matrix order 3

Therefore, $

\Rightarrow {\text{I}} = \dfrac{{5{\text{I}}}}{{|A|}} \\

\Rightarrow |{\text{A}}|I = 5I \\

$

On comparing the above equation, we get

Determinant of the matrix A, $|{\text{A}}| = 5$

Using the identity, \[|{\text{adjA}}| = {\left[ {|A|} \right]^{n - 1}}\] where n is the order of the matrix of A

Put $|{\text{A}}| = 5$ and ${\text{n}} = 3$ in the above identity, we have

\[ \Rightarrow |{\text{adjA}}| = {\left[ 5 \right]^{3 - 1}} = {5^2} = 25\]

Therefore, the determinant of matrix adjA is 25.

Option B is correct.

Note- Here, the inverse matrix only exists for non-singular matrices (i.e., determinant of that matrix whose inverse is required should always be non-zero). Also if in an equation two matrices are equal to each other then, order of both the matrices will be equal.

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