Animated Solution for Mathematics - Matrices and Determinants: If $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then which one of the following holds for all $n\ge 1$, by the principle of mathematical induction

Unveiling the Power of $A^n$

Welcome, future engineers. Today, we are going to demystify a classic JEE Advanced problem. Matrices are more than just grids of numbers; they are the language of transformations.

When we look at a matrix like $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$, we are looking at a specific linear transformation. Our goal is to find a general expression for $A^n$, the matrix raised to the power of $n$. This is a journey of pattern recognition, verification, and mathematical elegance.

The Pattern Hunt

Before we dive into heavy algebra, let us observe the behavior of $A$. We start by calculating the first few powers. We know $A^1=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$.

Now, let us compute $A^2=A\cdot A$. Using the standard row-by-column multiplication rule, we calculate:

Notice the change? Only the bottom-left element changed from $1$ to $2$. Let us push further to $A^3=A^2\cdot A$:

The Generalization

Do you see the beauty here? The pattern is screaming at us. For $n=1$, the bottom-left is $1$. For $n=2$, it is $2$. For $n=3$, it is $3$.

The other elements remain $1,0,1$. We can confidently generalize that for any $n\ge 1$, the matrix $A^n$ is:

This is the core geometric reality of our transformation. Now, we must verify which of the given options matches this result.

The Verification

We need to check if $A^n=nA-(n-1)I$. Let us perform the scalar multiplication and subtraction. First, $nA$:

Next, $(n-1)I$:

Finally, the subtraction:

The result is exactly $\left[\begin{array}{cc}1& 0\\ n& 1\end{array}\right]$, which matches our generalized $A^n$. We have successfully proven that $A^n=nA-(n-1)I$.

This problem teaches us that even in complex matrix algebra, observation and pattern recognition are your greatest allies. Keep practicing, stay curious, and remember that every matrix has a story to tell.