Question 1.11.1: LetM2×2 denote the collection of all 2×2 matrices with real ...

Let

A =\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}  and B=\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}

By the definition of matrix addition,

A+B=\begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{21} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}

and thus we see that A+B is also a 2×2 matrix. Recall that it only makes sense for matrices of the same size to be added; here we are simply pointing out the obvious fact that the sum of two matrices of the same size is yet another matrix of the same size. In the same way,

cA =\begin{bmatrix} ca_{11} & ca_{12} \\ ca_{21} & ca_{22} \end{bmatrix}

which shows that not only is the scalar multiple defined, but also that cA is a 2×2 matrix. Finally, if we let Z be the 2×2 matrix all of whose entries are zero,

Z =\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

en our work with matrix sums shows us immediately that A+Z =A for every possible 2×2 matrix A.