Question

If A is matrix of order 3, such that \[A\,\left( adj\text{ }A \right)\text{ }=\text{ }10\left( I \right),\] then what will be the value of \[|adj\text{ }A|\] ?
Choose the correct option.
A. 10
B . \[10\left( I \right)\]
C. 1
D . 100

Answer 1

Hint: Use the formula \[A\,(adj\text{ }A)\text{ }=\text{ }|A|\left( I \right)\] and \[|A|\,\times |(adj\text{ }A)|\text{ }=\text{ }|A{{|}^{n}}\], where n is the order of the matrix A.
Complete step-by-step answer:
In the question, we have to find the value of \[|adj\text{ }A|\]
Now, it is given that \[A\,\left( adj\text{ }A \right)\text{ }=\text{ }10\left( I \right)\] and we know that \[A\,(adj\text{ }A)\text{ }=\text{ }|A|\left( I \right)\].
So comparing the two we have:
\[\begin{align}
  & \Rightarrow A\,(adj\text{ }A)\text{ }=\text{ }|A|\left( I \right) \\
 & \Rightarrow A\,\left( adj\text{ }A \right)\text{ }=\text{ }10\left( I \right) \\
 & \Rightarrow |A|=10 \\
\end{align}\]
Now, we also know that:
\[|A|\,\times |(adj\text{ }A)|\text{ }=\text{ }|A{{|}^{n}}\]
Where n is given as 3, since n is the order of the matrix A.
So solving for \[|adj\text{ }A|\], we have:
\[\begin{align}
  & \Rightarrow |A|\,\times |(adj\text{ }A)|\text{ }=\text{ }|A{{|}^{n}} \\
 & \Rightarrow 10\,\times |(adj\text{ }A)|\text{ }=\text{ }{{10}^{3}} \\
 & \Rightarrow |(adj\text{ }A)|\text{ }=\text{ }{{10}^{2}} \\
 & \Rightarrow |(adj\text{ }A)|\text{ }=\text{ }100 \\
\end{align}\]
So, the required value of \[|adj\text{ }A|=100\] and hence the correct answer is option D.

Note: Here students should know one important property of determinant, that is we can find the determinant of only the square matrix. The non-square matrix will not have the determinant.