CUET PG 2021 — Mathematics PYQ

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Question

CUET PG 2021 — Mathematics PYQ

CUET PG | Mathematics | 2021

If $A = (a b am p; b am p; a) and A^{2} = (α β am p; β am p; α)$

Choose the correct answer:

Explanation

1. Matrix Multiplication

Given:

A = \begin{pmatrix} a & b \\ b & a \end{pmatrix}

We calculate $A^2$ as $A \times A$ :

A^2 = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \begin{pmatrix} a & b \\ b & a \end{pmatrix}

2. Performing the Multiplication

To multiply the matrices, we multiply the rows of the first matrix by the columns of the second:

Top-left element: $(a \times a) + (b \times b) = a^2 + b^2$
Top-right element: $(a \times b) + (b \times a) = ab + ba = 2ab$
Bottom-left element: $(b \times a) + (a \times b) = ba + ab = 2ab$
Bottom-right element: $(b \times b) + (a \times a) = b^2 + a^2$

So, the result is:

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

3. Comparing with the Given Matrix

We are given:

A^2 = \begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix}

By comparing the corresponding elements of the two matrices, we get:

\alpha = a^2 + b^2

\beta = 2ab

Final Result:

The values of $\alpha$ and $\beta$ are:

$\alpha = a^2 + b^2$
$\beta = 2ab$