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#matrix-columns

1 question from Linear Algebra

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Question 1: Single choice

mediumAI-checkedLinear Algebra

$\mathbf{e}_1 = (1, 0)^\top$ and $\mathbf{e}_2 = (0, 1)^\top$ are orthogonal. For which $2 \times 2$ matrices $A$ are $A\mathbf{e}_1$ and $A\mathbf{e}_2$ orthogonal too?

Those whose two columns are orthogonal, i.e. $A^\top A$ is diagonal1
Only the orthogonal matrices, i.e. those with $A^\top A = \mathbf{I}$ exactly2
The symmetric matrices, i.e. $A = A^\top$ 3
All invertible matrices, i.e. $\det A \neq 0$ 4

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