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Consider the set of all 2 x 2 matrices where the product of the elements on the main diagonal is zero.

$$M = \left \lbrace\begin{bmatrix}a&b\\c&d\end{bmatrix} \left. \right | a, b, c, d \in \mathbb R,ad = 0 \right \rbrace$$

Define addition and scalar multiplication on $M$ in the usual way.

Show that $M$, with respect to these operations of addition and scalar multiplication, is not a vector space by showing that one of the vector space axioms does not hold.

To be honest, I don't know how to perform addition and scalar multiplication to solve this question. If anyone could explain exactly how to solve this, that would be great!

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  • $\begingroup$ Is the set closed under addition? $\endgroup$ – Tpofofn Feb 12 '14 at 11:36
  • $\begingroup$ @Tpofofn yeah it is $\endgroup$ – Raba Feb 12 '14 at 11:37
  • $\begingroup$ @Raba are you sure about that? $\endgroup$ – Casteels Feb 12 '14 at 11:39
  • $\begingroup$ So $\begin{bmatrix}0&b\\c&d\end{bmatrix}$ and $\begin{bmatrix}a&b\\c&0\end{bmatrix}$ are in the set. Is their sum in the set? $\endgroup$ – Tpofofn Feb 12 '14 at 11:39
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    $\begingroup$ "You were told so"?? Since when that claiming things in mathematics gets reinforced by "being told so"?? $\endgroup$ – DonAntonio Feb 12 '14 at 12:36
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Let $A=\left( \begin{array}{cc} 0 & 1 \\ 1 & 1 \\ \end{array} \right) $ and $B=\left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \\ \end{array} \right) $ be two matrices in M. Then $$A+B=\left( \begin{array}{cc} 1 & 2 \\ 2 & 1 \\ \end{array} \right) $$ does not belong in M.

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