Animated Solution for Mathematics - Matrices and Determinants: If $A=\left(\begin{array}{ccc}1& 1& 2\\ 1& 3& 4\\ 1& -1& 3\end{array}\right)$, $B=adj(A)$ and $C=3A$ then $\frac{|adjB|}{|C|}$ is

Welcome, fellow explorer of the mathematical universe. Today, we are not just solving a matrix problem; we are uncovering the hidden architecture of linear transformations.

When you look at a matrix like it is easy to see it as a rigid grid of numbers. But in the world of JEE Advanced, we see it as a dynamic entity.

We are tasked with finding the ratio $\frac{|adj(B)|}{|C|}$, where $B=adj(A)$ and $C=3A$. While the brute-force approach of calculating the adjoint matrix $B$ would lead you into a labyrinth of tedious arithmetic, we will instead use the elegant shortcuts provided by the properties of determinants.

The Foundation - Calculating $|A|$

Everything in this problem rests on the determinant of our base matrix $A$. Think of the determinant as the 'scaling factor' of the transformation represented by the matrix.

To find $|A|$, we expand along the first row:

Calculating the minors:

Putting it all together: With $|A|=6$ in our pocket, we are ready to conquer the rest.

The Adjoint Property - Avoiding the Trap

Now, we need the determinant of $B$, where $B=adj(A)$. Do not reach for your pen to calculate the adjoint matrix!

Instead, remember the golden rule: for any $n\times n$ matrix, $|adj(A)|=|A{|}^{n-1}$. Since our matrix is $3\times 3$, $n=3$, so $|adj(A)|=|A{|}^{3-1}=|A{|}^2$.

Thus, $|B|=6^2=36$. But the problem asks for $|adj(B)|$. We apply the same property again: Since $|B|=36$, we have $|adj(B)|=36^2=1296$. See how we bypassed the entire process of finding the adjoint matrix? That is the power of conceptual mastery.

The Scalar Trap - The JEE Classic

Next, we tackle the denominator: $|C|=|3A|$. This is where many students stumble.

They pull out the $3$ and write $3|A|$. But remember, a determinant is not a simple matrix. When you multiply a matrix by a scalar $k$, you are multiplying every row by $k$.

In a $3\times 3$ matrix, you have three rows, so you pull out $k$ three times. The rule is $|kA|=k^n|A|$. For our $3\times 3$ matrix: Substituting our value for $|A|$, we get $|C|=27\times 6=162$.

The Final Synthesis

We have arrived at the final stage of our journey. We have the numerator, $|adj(B)|=1296$, and the denominator, $|C|=162$.

The ratio we seek is:

Let us perform this division with confidence. $162\times 8=1296$.

The ratio is exactly 8. You have successfully navigated the traps of matrix properties and scalar multiplication. Remember, in JEE Advanced, the most elegant path is almost always the right one.