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GATE 2023 AG – Question 36

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

$A=\begin{pmatrix}1&-1\\2&-1\end{pmatrix}$,$B=\begin{pmatrix}a&1\\b&-1\end{pmatrix}$ and $(A+B)^2=A^2+B^2$. Find a, b.

  1. a=4,b=1
  2. a=1,b=4
  3. a=0,b=4
  4. a=2,b=4

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Correct answer: (B) a=1,b=4

Explanation

Expanding the matrix square shows $AB+BA=0$. Its entries are $(2a+2-b,1-a;2a-2,4-2b)$. The off-diagonal equations give a=1,and either diagonal then gives b=4. Thus **B**. Cross-products must be retained because matrix multiplication is not commutative.

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