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Consider the vector space $M_{2\times2}$. Its vectors are $2 \times 2$ matrices with real entries. The addition operation is addition of matrices, and scalar multiplication is scalar multiplication of matrices.
(a) What is the zero vector of this vector space?
(b) Is the set of all $2 \times 2$ diagonal matrices a subspace of $M_{2\times2}$?
(c) Is the set of all $2 \times 2$ invertible matrices a subspace of $M_{2\times2}$?

What does this question mean by diagonal matrices?

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    $\begingroup$ Diagonal; matrices have $a_{12}=a_{21}=0$. $\endgroup$ Commented Dec 17, 2020 at 0:16
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    $\begingroup$ A matrix of the form $\begin{bmatrix}a&0\\0&b\end{bmatrix}$ $\endgroup$
    – saulspatz
    Commented Dec 17, 2020 at 0:17
  • $\begingroup$ I don't understand the votes to close for "Missing context or other details". The question is about a standard definition. Closure+comments suggesting they look in their notes (or some other interaction) would be helpful. Closing the question without comment is actively unhelpful. $\endgroup$
    – user1729
    Commented May 19, 2021 at 13:46

1 Answer 1

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When you are dealing with square matrices of order n (i.e. with n rows and n columns), a matrix $A=(a_{ij})$ is said to be diagonal if the entry $a_{ij}$ is zero for all indices $i\neq j$. Thus only the diagonal entries (that's how we call the entries $a_{ij}$ with i=j) can be nonzero.

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