Animated Solution for Mathematics - Matrices and Determinants: If $A=\left[\begin{array}{cc}\sqrt{2}& 1\\ -1& \sqrt{2}\end{array}\right]$, $B=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$, $C=AB{A}^T$ and $X=A^T{C}^{2}A$, then $\det X$ is equal to :

Imagine you are staring at a complex matrix equation. You see $A$, you see $B$, and then you see $C=AB{A}^T$ and $X=A^T{C}^{2}A$.

Your first instinct might be to start multiplying, to dive into the rows and columns, and to perform a massive amount of arithmetic. Stop. Take a breath.

In the world of JEE Advanced, the most elegant path is rarely the one that requires the most writing. Today, we are going to solve for $\det (X)$ without ever performing a full matrix multiplication.

First, let us look at our building blocks. We have:

To find the determinant of a $2\times 2$ matrix $M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, we use the classic formula $\det (M)=ad-bc$.

For matrix $A$, this is $(\sqrt{2})(\sqrt{2})-(1)(-1)$, which simplifies to $2-(-1)=3$. For matrix $B$, it is even simpler: $(1)(1)-(0)(1)=1$.

We have our values: $\det (A)=3$ and $\det (B)=1$.

Now, we enter the realm of matrix properties. This is where the JEE examiners test your conceptual depth.

We know that $\det (MN)=\det (M)\cdot \det (N)$. We also know that $\det (A^T)=\det (A)$.

With these tools, we can dismantle the expression for $C=AB{A}^T$. Instead of calculating $C$, we calculate its determinant directly:

Substituting our known values, we get $\det (C)=3\cdot 1\cdot 3=9$. See how the complexity just melted away?

Finally, we tackle $X=A^T{C}^{2}A$. Applying the same property, we get:

We know $\det (C^2)=(\det C{)}^2$, so $\det (C^2)=9^2=81$.

Now, we just plug everything in: $\det (X)=3\cdot 81\cdot 3$. That is $9\cdot 81$, which equals 729.

We have arrived at the answer, not through brute force, but through the elegant application of mathematical laws. Remember, in mathematics, the most powerful tool is often the one that lets you see the structure before you start the calculation.