If M = [[5/2, 3/2], [-3/2, -1/2]], then which of the following matrices is equal to M²⁰²²? (A) [[303

Q: Wait, why can't we just multiply the matrix 2022 times? Is there a shortcut?

In the JEE Advanced arena, time is your most precious currency. If you attempt to multiply a matrix 2022 times, you will not only run out of time but also succumb to calculation errors. The problem is designed to test your ability to spot a pattern, not your endurance for arithmetic. Always look for the 'hidden' structure!

Q: Why can we use the Binomial Theorem here? Isn't matrix multiplication non-commutative?

You are absolutely right to be cautious! Matrix multiplication is generally non-commutative ($AB \neq BA$). However, the Binomial Theorem $(A+B)^n$ works if and only if $AB = BA$. In our case, we are expanding $(I + \frac{3}{2}A)^n$. Since the identity matrix $I$ commutes with every matrix ($IA = AI = A$), the condition is satisfied perfectly. You are safe to proceed!

Q: What exactly is a 'Nilpotent' matrix and why does it make the math vanish?

A nilpotent matrix is a matrix that, when raised to some power $k$, becomes the zero matrix ($O$). In our case, $A^2 = O$. This is a mathematical superpower! It means that in the binomial expansion, every term involving $A^2, A^3, \dots$ becomes zero. It effectively 'kills' the infinite series, leaving you with only the first two terms. It is the ultimate simplification tool.

Question

If $M = \begin{pmatrix} \frac{5}{2} & \frac{3}{2} \ -\frac{3}{2} & -\frac{1}{2} \end{pmatrix}$, then which of the following matrices is equal to $M^{2022}$?

Options:

(A) $\begin{pmatrix} 3034 & 3033 \ -3033 & -3032 \end{pmatrix}$

(B) $\begin{pmatrix} 3034 & -3033 \ 3033 & -3032 \end{pmatrix}$

(C) $\begin{pmatrix} 3033 & 3032 \ -3032 & -3031 \end{pmatrix}$

(D) $\begin{pmatrix} 3032 & 3031 \ -3031 & -3030 \end{pmatrix}$

Sigma Percentile · Accepted Answer

### The Illusion of Complexity: Facing the Giant

Imagine you are standing before a massive, imposing mountain. The problem asks you to calculate $M^{2022}$, where:

$$M = \begin{pmatrix} \frac{5}{2} & \frac{3}{2} \ -\frac{3}{2} & -\frac{1}{2} \end{pmatrix}$$

If you look at this and think, "I need to multiply this matrix by itself 2022 times," you are falling into the trap. The JEE Advanced examiners are not testing your ability to perform tedious multiplication; they are testing your ability to see the hidden elegance beneath the surface.

### Phase 1: The Art of Decomposition

Whenever you see a matrix with numbers that look "almost" like the identity matrix, your alarm bells should ring. Look at the diagonal elements: $\frac{5}{2}$ and $-\frac{1}{2}$. They are just a small step away from $1$.

Let us rewrite them:

$$\frac{5}{2} = 1 + \frac{3}{2}$$

$$-\frac{1}{2} = 1 - \frac{3}{2}$$

By doing this, we can peel away the identity matrix $I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$ like the skin of an onion. We are left with:

$$M = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} + \begin{pmatrix} \frac{3}{2} & \frac{3}{2} \ -\frac{3}{2} & -\frac{3}{2} \end{pmatrix}$$

Now, factor out that $\frac{3}{2}$ from the second matrix. We define a new matrix $A = \begin{pmatrix} 1 & 1 \ -1 & -1 \end{pmatrix}$. Suddenly, our terrifying matrix $M$ has transformed into the elegant form: $M = I + \frac{3}{2}A$.

### Phase 2: The Nilpotent Discovery

Why did we isolate $A$? Because $A$ is special. Let us test its power by squaring it:

$$A^2 = \begin{pmatrix} 1 & 1 \ -1 & -1 \end{pmatrix} \begin{pmatrix} 1 & 1 \ -1 & -1 \end{pmatrix} = \begin{pmatrix} (1)(1) + (1)(-1) & (1)(1) + (1)(-1) \ (-1)(1) + (-1)(-1) & (-1)(1) + (-1)(-1) \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}$$

It is the null matrix! This is what mathematicians call a **Nilpotent** matrix of index 2. This is a gift, as it means $A^2 = O$, $A^3 = O$, and so on.

### Phase 3: The Binomial Shortcut

Now, we return to our goal: $M^{2022} = (I + \frac{3}{2}A)^{2022}$. Because $I$ and $A$ commute, we can use the Binomial Theorem:

$$(I + \frac{3}{2}A)^{2022} = I^{2022} + \binom{2022}{1} I^{2021} \left(\frac{3}{2}Aight) + \binom{2022}{2} I^{2020} \left(\frac{3}{2}Aight)^2 + \dots$$

Look at the terms after the second one. They all contain $A^2, A^3, \dots$, which we just proved are all zero. The infinite series collapses into just two terms:

$$M^{2022} = I + 2022 \cdot \left(\frac{3}{2}Aight)$$

Calculating the scalar: $2022 \cdot \frac{3}{2} = 1011 \cdot 3 = 3033$. So, $M^{2022} = I + 3033A$.

### The Final Victory

We are almost there. Substitute the matrices back in:

$$M^{2022} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} + 3033 \begin{pmatrix} 1 & 1 \ -1 & -1 \end{pmatrix} = \begin{pmatrix} 1+3033 & 0+3033 \ 0-3033 & 1-3033 \end{pmatrix}$$

The final result is:

$$\mathbf{M^{2022} = \begin{pmatrix} 3034 & 3033 \ -3033 & -3032 \end{pmatrix}}$$

Look at that result. It is clean, precise, and beautiful. You didn't need to multiply 2022 matrices; you just needed to understand the soul of the matrix.

Animated Solution for Mathematics - Matrices and Determinants: If $M = (\frac{5}{2} - \frac{3}{2} \frac{3}{2} - \frac{1}{2})$ , then which of the following matrices is equal to $M^{2022}$ ?

Select Answer:

Visualized Solution

Decomposing Matrix $M$

Extracting Identity Matrix $I$

Defining Matrix $A$

Investigating Matrix $A$

Computing $A^{2}$

$A$ is a Nilpotent Matrix

Setup for $M^{2022}$

Applying Binomial Theorem

Simplifying the Expression

Substituting Matrices Back

Final Matrix Addition

The Sigma Insight: Algebraic Operations on Matrices

The Illusion of Complexity

Phase 1

Phase 2

Phase 3

The Final Victory

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Decomposing Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } M

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Investigating Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A is a Nilpotent Matrix

Setup for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } M2022