Question
Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
We need to calculate $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))|$ given $|A| = 6$ for a $3 \times 3$ matrix $A$. Let $p=3$ be the dimension of the matrix.
Key Matrix Properties Used
- $|\text{adj}(M)| = |M|^{p-1}$
- $|kA| = k^p |A|$
- $|AB| = |A||B|$
- $|A^k| = |A|^k$
Step-by-Step Calculation
- Let the expression be $E = |\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))|$.
- Using property $|\text{adj}(M)| = |M|^{p-1}$, we get $E = |3\text{adj}(A^2 \cdot \text{adj}(2A))|^{p-1} = |3\text{adj}(A^2 \cdot \text{adj}(2A))|^2$.
- Let $X = A^2 \cdot \text{adj}(2A)$. Then $E = |3\text{adj}(X)|^2$.
- Using property $|kA| = k^p |A|$, we get $|3\text{adj}(X)| = 3^p |\text{adj}(X)| = 3^3 |\text{adj}(X)| = 27 |\text{adj}(X)|$.
- Substituting this back into the expression for $E$: $E = (27 |\text{adj}(X)|)^2 = 27^2 |\text{adj}(X)|^2$.
- Using property $|\text{adj}(M)| = |M|^{p-1}$ again for $|\text{adj}(X)|$: $|\text{adj}(X)| = |X|^{p-1} = |X|^2$.
- Therefore, $E = 27^2 (|X|^2)^2 = 27^2 |X|^4$.
- Now, we need to find $|X| = |A^2 \cdot \text{adj}(2A)|$. Using property $|AB| = |A||B|$, we get $|X| = |A^2| \cdot |\text{adj}(2A)|$.
- Calculate $|A^2|$: Using property $|A^k| = |A|^k$, $|A^2| = |A|^2 = 6^2 = 36$.
- Calculate $|\text{adj}(2A)|$: Using properties $|\text{adj}(M)| = |M|^{p-1}$ and $|kA| = k^p |A|$, we get $|\text{adj}(2A)| = |2A|^{p-1} = |2A|^2$. First, $|2A| = 2^p |A| = 2^3 |A| = 8 \cdot 6 = 48$. So, $|\text{adj}(2A)| = (48)^2$.
- Now, find $|X|$: $|X| = 36 \cdot (48)^2$.
- Substitute $|X|$ back into the expression for $E$: $E = 27^2 \cdot (36 \cdot (48)^2)^4$.
- Express terms using prime factorization: $27 = 3^3$, $36 = 2^2 \cdot 3^2$, $48 = 2^4 \cdot 3$.
- $E = (3^3)^2 \cdot ((2^2 \cdot 3^2) \cdot (2^4 \cdot 3)^2)^4$ $E = 3^6 \cdot ((2^2 \cdot 3^2) \cdot (2^8 \cdot 3^2))^4$ $E = 3^6 \cdot (2^{2+8} \cdot 3^{2+2})^4$ $E = 3^6 \cdot (2^{10} \cdot 3^4)^4$ $E = 3^6 \cdot 2^{40} \cdot 3^{16}$ $E = 2^{40} \cdot 3^{6+16}$ $E = 2^{40} \cdot 3^{22}$.
- The result is given in the form $2^m \cdot 3^n$. Comparing, we have $m = 40$ and $n = 22$.
- We need to find $m+n$. $m+n = 40 + 22 = 62$.
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