JEE Mathematics 2026 Question with Solution

Given: $|A| = 6$ and $A$ is a $3 \times 3$ matrix.

Find: $m+n$ if

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

Let

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

For a $3 \times 3$ matrix, we use

$$|\operatorname{adj}(B)| = |B|^{3-1} = |B|^2$$

Hence,

$$|\operatorname{adj}(\operatorname{adj}(B))| = |\operatorname{adj}(B)|^2 = (|B|^2)^2 = |B|^4$$

Now,

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

Also,

$$|A^2| = |A|^2$$

and

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

Since $A$ is of order $3$,

$$|2A| = 2^3 |A| = 8|A|$$

Therefore,

$$|B| = |A|^2 \cdot (8|A|)^2$$ $$= |A|^2 \cdot 64|A|^2$$ $$= 64|A|^4$$

Substituting $|A|=6$,

$$|B| = 64 \cdot 6^4$$

Now,

$$64 = 2^6, \qquad 6^4 = (2\cdot 3)^4 = 2^4 \cdot 3^4$$

So,

$$|B| = 2^6 \cdot 2^4 \cdot 3^4 = 2^{10} \cdot 3^4$$

Hence,

$$|\operatorname{adj}(\operatorname{adj}(B))| = |B|^4 = (2^{10} \cdot 3^4)^4 = 2^{40} \cdot 3^{16}$$

Therefore, $m=40$ and $n=16$, so the required sum is $56$.

The correct numerical answer is $56$. The answer key showed $62$, but the solution working gives $56$, so $56$ is taken as correct.

Given: $|A|=6$ for a $3 \times 3$ matrix.

Find: $m+n$.

The key determinant facts used are:

1. $|XY| = |X|\,|Y|$
2. $|A^2| = |A|^2$
3. For a $3 \times 3$ matrix, $|\operatorname{adj}(M)| = |M|^2$
4. For a scalar multiple of a $3 \times 3$ matrix, $|kA| = k^3 |A|$

Take

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

Then

$$|\operatorname{adj}(\operatorname{adj}(B))| = |\operatorname{adj}(B)|^2$$

because $\operatorname{adj}(B)$ is also a $3 \times 3$ matrix. Also,

$$|\operatorname{adj}(B)| = |B|^2$$

Therefore,

$$|\operatorname{adj}(\operatorname{adj}(B))| = (|B|^2)^2 = |B|^4$$

Now evaluate $|B|$:

$$|B| = |A^2|\,|\operatorname{adj}(2A)|$$ $$= |A|^2 \cdot |2A|^2$$

Since

$$|2A| = 2^3|A| = 8|A|$$

we get

$$|B| = |A|^2 \cdot (8|A|)^2 = 64|A|^4$$

Substitute $|A|=6$:

$$|B| = 64 \cdot 6^4$$

Expressing in powers of $2$ and $3$,

$$64 \cdot 6^4 = 2^6 \cdot (2\cdot 3)^4 = 2^6 \cdot 2^4 \cdot 3^4 = 2^{10} \cdot 3^4$$

Thus,

$$|\operatorname{adj}(\operatorname{adj}(B))| = (2^{10} \cdot 3^4)^4 = 2^{40} \cdot 3^{16}$$

So,

$$m=40, \qquad n=16$$

Hence,

$$m+n = 56$$

Therefore, the answer is $56$.