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Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to (A) | A | (B) | A |^2 (C) | A |^3 (D)3| A |

Q : 17        Let A be a nonsingular square matrix of order $\small 3\times 3$. Then $\small |adjA|$ is equal to

(A) $\small |A|$      (B) $\small |A|^2$      (C) $\small |A|^3$      (D) $\small 3|A|$

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We know the identity $(adjA)A = |A| I$

Hence we can determine the value of $|(adjA)|$.

Taking both sides determinant value we get,

$|(adjA)A| = ||A| I|$     or   $|(adjA)||A| = ||A||| I|$

or taking R.H.S.,

$||A||| I| = \begin{vmatrix} |A| & 0&0 \\ 0&|A| &0 \\ 0&0 &|A| \end{vmatrix}$

$= |A| (|A|^2) = |A|^3$

or, we have then $|(adjA)||A| = |A|^3$

Therefore $|(adjA)| = |A|^2$

Hence the correct answer is B.

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