Question

Let \(A\) be an invertible matrix. Identify the correct statement(s) from the following.
(I) \(A(\text{adj }A^{-1})=\frac{1}{|A|}A^{2}\) (II) \((\text{adj }A^{-1})^{-1}=\frac{1}{|A|}A^{-1}\) (III) \((\text{adj }A)=\frac{1}{|A|}A^{-1}\)

• A. only (I)
• B. only (I) and (II)
• C. only (III)
• D. only (II) and (III)

Hint:

Use adj(A inverse) = A divided by |A| and adj A = |A| times A inverse to test each statement.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
For an invertible matrix, \(\text{adj }A=|A|A^{-1}\) and \(|A^{-1}|=\dfrac{1}{|A|}\).

Step 2: Statement I
\(\text{adj }A^{-1}=|A^{-1}|\,(A^{-1})^{-1}=\dfrac{A}{|A|}\).
Then \(A(\text{adj }A^{-1})=\dfrac{A^{2}}{|A|}\), so (I) is true.

Step 3: Statements II and III
\((\text{adj }A^{-1})^{-1}=\left(\dfrac{A}{|A|}\right)^{-1}=|A|A^{-1}\), not \(\dfrac{1}{|A|}A^{-1}\), so (II) is false.
\(\text{adj }A=|A|A^{-1}\), not \(\dfrac{1}{|A|}A^{-1}\), so (III) is false.

Step 4: Compare the options
Only statement (I) is correct.

Final Answer:
Only statement (I) is correct. \[ \boxed{\text{(A) }\text{only (I)}} \]