{pmatrix} a & -b \\ b & a {pmatrix} × {pmatrix} -a & b \\ b & a {pmatrix}

Correct Answer is : (−a2−b200a2−b2){pmatrix} -a²-b² & 0 \\ 0 & a²-b² {pmatrix}(−a2−b200a2−b2)

{pmatrix} a²+b² & 0 \\ 0 & a²+b² {pmatrix}

{pmatrix} -a²-b² & 0 \\ 0 & a²-b² {pmatrix}

{pmatrix} a²-b² & 0 \\ 0 & a²+b² {pmatrix}

{pmatrix} a²-b² & 0 \\ 0 & a²-b² {pmatrix}

Correct Answer is : (−a2−b200a2−b2){pmatrix} -a²-b² & 0 \\ 0 & a²-b² {pmatrix}(−a2−b200a2−b2)

Test Index

Matrices

Question : 35 of 36

Marks: +1, -0

\begin{pmatrix} a & -b \\ b & a \end{pmatrix} \times \begin{pmatrix} -a & b \\ b & a \end{pmatrix}

Solution:

Concept:

Perform matrix multiplication of two 2×2 matrices.

Explanation:

Let

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

and

B = \begin{pmatrix} -a & b \\ b & a \end{pmatrix}

Compute each element of the product

AB

(AB)_{11} = a(-a) + (-b)b = -a^2 - b^2

(AB)_{12} = a(b) + (-b)a = ab - ab = 0

(AB)_{21} = b(-a) + a(b) = -ab + ab = 0

(AB)_{22} = b(b) + a(a) = b^2 + a^2 = a^2 + b^2

Thus,

AB = \begin{pmatrix} -a^2-b^2 & 0 \\ 0 & a^2+b^2 \end{pmatrix}

Answer:

The result is

\begin{pmatrix} -a^2-b^2 & 0 \\ 0 & a^2+b^2 \end{pmatrix}

, which matches option B if we note that option B has top-left

-a^2-b^2

(the bottom-right is misprinted as

a^2-b^2

but the intended answer is B).

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