If {A} = {pmatrix} -2 & 2 \\ 2 & -2 {pmatrix}, then which one of the following is correct?

Correct Answer is : A2=−4AA² = - 4AA2=−4A

Correct Answer is : A2=−4AA² = - 4AA2=−4A

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UPSC NDA 2 2017 Mathematics Solved Paper

Question : 98 of 120

Marks: +1, -0

\mathrm{A} = \begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix}

, then which one of the following is correct?

Solution:

Concept:

Matrix multiplication and scalar multiplication.

Explanation:

First, compute

A^2

by multiplying matrix

A

with itself.

A = \begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix}

A^2 = A \times A = \begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix}

Multiply each row of the first matrix by each column of the second:

Element (1,1):

(-2)(-2) + (2)(2) = 4 + 4 = 8

Element (1,2):

(-2)(2) + (2)(-2) = -4 - 4 = -8

Element (2,1):

(2)(-2) + (-2)(2) = -4 - 4 = -8

Element (2,2):

(2)(2) + (-2)(-2) = 4 + 4 = 8

A^2 = \begin{pmatrix} 8 & -8 \\ -8 & 8 \end{pmatrix}

Now compute

-4A

-4A = -4 \times \begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix} = \begin{pmatrix} 8 & -8 \\ -8 & 8 \end{pmatrix}

Thus

A^2 = -4A

Answer:

Option B:

A^{2} = -4A

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