Animated Solution for Mathematics - Matrices and Determinants: If the matrices $A=\left[\begin{array}{ccc}1& 1& 2\\ 1& 3& 4\\ 1& -1& 3\end{array}\right]$, $B=\text{adj }A$ and $C=3A$, then $\frac{|\text{adj }B|}{|C|}$ is equal to:

Imagine you are standing before a complex matrix $A=\left[\begin{array}{ccc}1& 1& 2\\ 1& 3& 4\\ 1& -1& 3\end{array}\right]$. The problem asks for the ratio $\frac{|\text{adj }B|}{|C|}$, where $B=\text{adj }A$ and $C=3A$.

In the world of JEE Advanced, brute force is often the path to a trap. Let us walk through this with elegance and precision by utilizing matrix properties.

First, we must identify the order of our matrix. Since $A$ is a $3\times 3$ matrix, we have $n=3$. This number is the key that unlocks all our properties.

Now, let us calculate the determinant of $A$. Expanding along the first row:

Simplifying this, we get:

With $|A|=6$ in our pocket, we are ready to conquer the rest.

We need the determinant of the adjoint of $B$, where $B=\text{adj }A$. The property $|\text{adj }M|=|M{|}^{n-1}$ is our best friend here.

First, let us find $|B|$. Since $B=\text{adj }A$:

Now, we need $|\text{adj }B|$. Applying the same property again:

We have successfully navigated the numerator without ever calculating a single cofactor.

Now for the denominator, $|C|=|3A|$. Remember the scalar property: $|kM|=k^n|M|$.

Here, $k=3$ and $n=3$. So:

We have our numerator, $1296$, and our denominator, $162$. The final ratio is:

Performing the division, we find the result is exactly 8. Through the power of properties, we bypassed the nightmare of matrix algebra and arrived at the truth.