If A is a square matrix of order 3 × 3 and |A| = 6, then which of the following is/are correct?
(A) |2 A| = 48
(B) |adj adj A| = 1296
(C) |adj A| = 216
(D) \( \mathrm{A}^{-1}=\displaystyle\frac{1}{|\mathrm{A}|} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
For a square matrix of order n:
\( \displaystyle |k\mathrm{A}|=k^n|\mathrm{A}| \)
\( \displaystyle |\mathrm{adj}\,\mathrm{A}|=|\mathrm{A}|^{n-1} \)
\( \displaystyle |\mathrm{adj}(\mathrm{adj}\,\mathrm{A})|=|\mathrm{A}|^{(n-1)^2} \)
Here n = 3 and |A| = 6:
\( \displaystyle |2\mathrm{A}|=2^3\times 6=48 \)
\( \displaystyle |\mathrm{adj}\,\mathrm{A}|=6^{2}=36 \)
\( \displaystyle |\mathrm{adj}(\mathrm{adj}\,\mathrm{A})|=6^{4}=1296 \)
So |2A| = 48, |adj A| = 36, |adj adj A| = 1296, and the inverse is the matrix \( \mathrm{A}^{-1}=\frac{1}{|\mathrm{A}|}\,\mathrm{adj}\,\mathrm{A} \).
Checking the statements
(A) |2 A| = 48 — Correct. \( 2^3\times 6=48 \).
(B) |adj adj A| = 1296 — Correct. \( 6^{(3-1)^2}=6^4=1296 \).
(C) |adj A| = 216 — Incorrect. \( |\mathrm{adj}\,\mathrm{A}|=6^{3-1}=36 \); 216 = 63 uses the power n instead of n − 1.
(D) \( \mathrm{A}^{-1}=\frac{1}{|\mathrm{A}|} \) — Incorrect. The inverse is the matrix \( \frac{1}{|\mathrm{A}|}\,\mathrm{adj}\,\mathrm{A} \); it is \( |\mathrm{A}^{-1}| \) that equals \( \frac{1}{|\mathrm{A}|} \).
Tip: Remember the powers for order n: kA gives \( k^n \), adj A gives \( n-1 \), adj(adj A) gives \( (n-1)^2 \).