Answer
Verified
482.7k+ views
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE