If A = [a b; b a] and A² = [α β; β α], then (a) α = 2ab, β = a² + b² (b) α = a² + b², β = ab (c) α =

Q: Wait, why can't I just write $A^2 = \begin{bmatrix} a^2 & b^2 \\ b^2 & a^2 \end{bmatrix}$? It seems so much faster!

Ah, the classic trap! My dear student, matrix multiplication is not element-wise multiplication. It is a linear combination of rows and columns. If you square the elements, you are ignoring the interaction between the rows and columns, which is the very soul of matrix algebra. Always stick to the row-by-column rule!

Q: Is it always true that $A^2$ will have the same structure as $A$?

Excellent observation! In this specific case, yes, because the matrix is symmetric and has a specific structure. However, do not assume this for every matrix. Always perform the multiplication to verify the resulting structure.

Q: Do I need to worry about the order of multiplication here?

In general, matrix multiplication is not commutative ($AB \neq BA$). However, for $A^2$, you are multiplying $A$ by itself, so the order doesn't change the result. But always be mindful of the order in more complex problems!

Question

If $A = \begin{bmatrix} a & b \ b & a \end{bmatrix}$ and $A^2 = \begin{bmatrix} \alpha & \beta \ \beta & \alpha \end{bmatrix}$, then

Options:

(A) $\alpha = 2ab, \beta = a^2 + b^2$

(B) $\alpha = a^2 + b^2, \beta = ab$

(C) $\alpha = a^2 + b^2, \beta = 2ab$

(D) $\alpha = a^2 + b^2, \beta = a^2 - b^2$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future IITians! Today, we are going to demystify a problem that often trips up even the brightest minds in the early stages of their JEE preparation.

We are dealing with a $2 	imes 2$ matrix defined as:

$$A = \begin{bmatrix} a & b \ b & a \end{bmatrix}$$

Our task is to find $A^2$ and identify the variables $\alpha$ and $\beta$ in the resulting matrix $A^2 = \begin{bmatrix} \alpha & \beta \ \beta & \alpha \end{bmatrix}$.

### The Trap: The Siren Song of Element-wise Squaring

Before we dive into the math, let's address the elephant in the room. When you see $A^2$, your brain might instinctively want to square every element inside the matrix, thinking it results in $\begin{bmatrix} a^2 & b^2 \ b^2 & a^2 \end{bmatrix}$.

Stop right there! This is the most common trap in matrix algebra.

Matrices are not just collections of numbers; they are operators. Squaring a matrix means multiplying it by itself, which involves a specific, rigorous process: the **row-by-column multiplication**.

### The Execution: The Row-by-Column Dance

To find $A^2$, we must perform the operation $A 	imes A$. Let's write it out clearly:

$$A^2 = \begin{bmatrix} a & b \ b & a \end{bmatrix} 	imes \begin{bmatrix} a & b \ b & a \end{bmatrix}$$

Now, we use the row-by-column rule. For the top-left element, we take the first row of the first matrix and the first column of the second matrix:

$$(a 	imes a) + (b 	imes b) = a^2 + b^2$$

Next, for the top-right element, we take the first row of the first matrix and the second column of the second matrix:

$$(a 	imes b) + (b 	imes a) = ab + ab = 2ab$$

For the bottom-left element, we take the second row of the first matrix and the first column of the second matrix:

$$(b 	imes a) + (a 	imes b) = ba + ab = 2ab$$

Finally, for the bottom-right element, we take the second row of the first matrix and the second column of the second matrix:

$$(b 	imes b) + (a 	imes a) = b^2 + a^2 = a^2 + b^2$$

### The Result: The Beauty of Symmetry

Look at what we have created! Our resulting matrix is:

$$A^2 = \begin{bmatrix} a^2 + b^2 & 2ab \ 2ab & a^2 + b^2 \end{bmatrix}$$

By comparing this to the given form $A^2 = \begin{bmatrix} \alpha & \beta \ \beta & \alpha \end{bmatrix}$, we can immediately identify our unknowns.

The top-left element tells us that **$\alpha = a^2 + b^2$**. The top-right element tells us that **$\beta = 2ab$**.

It is elegant, it is symmetric, and it is correct. Remember, in JEE Advanced, the beauty lies in the process. Never rush to the answer; enjoy the dance of the rows and columns.

Animated Solution for Mathematics - Matrices and Determinants: If $A = [a b b a]$ and $A^{2} = [α β β α]$ , then

Select Answer:

Visualized Solution

The Given Matrix $A$

The Matrix Squaring Rule

Setting up $A \times A$

Calculating Top-Left Element

Calculating Top-Right Element

Calculating Bottom-Left Element

Calculating Bottom-Right Element

Comparing with Given $A^{2}$

Finding $α$

Finding $β$

The Sigma Insight: Algebraic Operations on Matrices

Analyzing the Setup

The Trap

The Execution

The Result

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Animated Solution for Mathematics - Matrices and Determinants: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=[ab​ba​] and A2=[αβ​βα​], then

Select Answer:

Visualized Solution

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

The Matrix Squaring Rule

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

Calculating Top-Left Element

Calculating Top-Right Element

Calculating Bottom-Left Element

Calculating Bottom-Right Element

Comparing with Given .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A2

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

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