JEE Mathematics 2023 Question with Solution

Step-by-step solution

Standard Method

Given: $A$ is a $3\times3$ matrix and $|A|=2$.

Find: $|3\operatorname{adj}(|3A|A^2)|$.

First, calculate

$$|3A|=3^3|A|=3^3\cdot2$$

So,

$$|3A|=2\cdot3^3$$

Now,

$$\operatorname{adj}(|3A|A^2)=\operatorname{adj}\{(2\cdot3^3)A^2\}$$

Using the property shown in the solution,

$$\operatorname{adj}(|3A|A^2)=(2\cdot3^3)^2\cdot(\operatorname{adj}A)^2$$

Hence,

$$=2^2\cdot3^6\cdot(\operatorname{adj}A)^2$$

Now evaluate

$$|3\operatorname{adj}(|3A|A^2)|$$

So,

$$|3\operatorname{adj}(|3A|A^2)|=|2^2\cdot3^7\cdot(\operatorname{adj}A)^2|$$

Therefore,

$$=(2^2\cdot3^7)^3\cdot|\operatorname{adj}A|^2$$

Also, for a $3\times3$ matrix,

$$|\operatorname{adj}A|=|A|^2$$

Thus,

$$=(2^2\cdot3^7)^3\cdot(|A|^2)^2$$

Substituting $|A|=2$,

$$=2^6\cdot3^{21}\cdot(2^2)^2$$ $$=2^{10}\cdot3^{21}$$

Now rewrite in terms of the options:

$$2^{10}\cdot3^{21}=3^{11}\cdot(2\cdot3)^{10}=3^{11}\cdot6^{10}$$

Therefore, the correct option is B.

Option Matching

From the working, the final value is

$$2^{10}\cdot3^{21}$$

To compare with the options, write

$$6^{10}=(2\cdot3)^{10}=2^{10}\cdot3^{10}$$

Hence,

$$2^{10}\cdot3^{21}=3^{11}\cdot2^{10}\cdot3^{10}=3^{11}\cdot6^{10}$$

So the expression matches option B exactly.

Common mistakes

• Using $|3A|=3|A|$ instead of $|3A|=3^3|A|$. For a $3\times3$ matrix, multiplying the matrix by $3$ multiplies the determinant by $3^3$. Always use $|kA|=k^n|A|$ for an $n\times n$ matrix.

• Using the wrong determinant formula for the adjugate. For an $n\times n$ matrix, $|\operatorname{adj}A|=|A|^{n-1}$, so here $|\operatorname{adj}A|=|A|^2$. Do not replace it by $|A|$ or $|A|^3$.

• Stopping at $2^{10}\cdot3^{21}$ and failing to compare with the given options. Since the options are written in powers of $3$ and $6$, rewrite $2^{10}\cdot3^{21}$ as $3^{11}\cdot6^{10}$ before choosing the answer.