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:

& \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{aligned}$$ Now, we also know that: $$|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{aligned} & \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{aligned}$$ So, the required value of $$|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.