# Let $T$ be an $m\times n$ matrix. Show that $T = -T$ if and only if $T$ is the $m \times n$ zero matrix. [closed]

This question has confused me a bit in a previous exam paper I am doing. It seems too obvious to prove. The question even looks like it proves itself to me.

This question is for 5 marks and I don't even know how to start to prove it.

## closed as off-topic by user99914, Jendrik Stelzner, Juniven, Shailesh, ancientmathematicianSep 11 '18 at 11:15

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• You can check all the entries of the matrix and found that they are zero. – user99914 Nov 2 '14 at 5:43

Hint :Every entry $x$ satisfies the equation $x=-x$
let the matrix $$T=\begin{pmatrix}a_{11} &a_{12}&a_{13}&\ldots&a_{1n}\\a_{21}&a_{22}&a_{23}&\ldots&a_{2n}\\.\\.\\.\\a_{m1}&a_{m2}&a_{m3}&\ldots&a_{mn}\end{pmatrix}_{m\times n}=\begin{pmatrix}a_{ij}\end{pmatrix}$$ then $$-T=\begin{pmatrix}-a_{11} &-a_{12}&-a_{13}&\ldots&-a_{1n}\\-a_{21}&-a_{22}&-a_{23}&\ldots&-a_{2n}\\.\\.\\.\\-a_{m1}&-a_{m2}&-a_{m3}&\ldots&-a_{mn}\end{pmatrix}_{m\times n}=\begin{pmatrix}-a_{ij}\end{pmatrix}$$ And if $T=-T$
$\therefore$ $$a_{ij}=-a_{ij}$$ which is possible if and only if $$a_{ij}=-a_{ij}=0$$ therefore $T$ is a zero matrix.