## Rules of matrix calculations.

Prove or disprove, whether for matrices $$\mathbf A,\mathbf B,\mathbf C$$ and $$\mathbf 0$$ of the same type and real numbers $$\alpha,\beta$$ holds:

 a) $$\mathbf A+(\mathbf B+\mathbf C)=(\mathbf A+\mathbf B)+\mathbf C$$ h) $$\alpha(\mathbf A+\mathbf B)=\alpha\mathbf A+\alpha\mathbf B$$ b) $$\mathbf A+\mathbf B=\mathbf B+\mathbf A$$ i) $$(\alpha+\beta)\mathbf A=\alpha\mathbf A+\beta\mathbf A$$ c) $$\mathbf A+\mathbf 0=\mathbf A$$ j) $$\alpha\mathbf A+\beta\mathbf B=(\alpha+\beta)(\mathbf A+\mathbf B)$$ d) $$\alpha(\beta\mathbf A)=(\alpha\beta)\mathbf A$$ k) $$(\mathbf A^T)^T=\mathbf A$$ e) $$\alpha(\beta\mathbf A)=\beta(\alpha\mathbf A)$$ l) $$(\mathbf A+\mathbf B)^T=\mathbf A^T+\mathbf B^T$$ f) $$\mathbf A+(-1)\mathbf A=\mathbf 0$$ m) $$(\alpha\mathbf A)^T=\alpha(\mathbf A^T)$$ g) $$1\mathbf A=\mathbf A$$
• #### Resolution

We compare matrices on th left and on the right side elementwise:

a) $$(\mathbf A+(\mathbf B+\mathbf C))_{ij}=a_{ij}+(\mathbf B+\mathbf C)_{ij}=a_{ij}+(b_{ij}+c_{ij})=(a_{ij}+b_{ij})+c_{ij}=(\mathbf A+\mathbf B)_{ij}+c_{ij}=((\mathbf A+\mathbf B)+\mathbf C)_{ij}$$

The first, second, fourth and fifth equalities are the definition of matrix sum. The third equation is the asociativity law of the addition of real numbers.

b-l) could be written accordingly, only j) does not hold, the correct variant is:
$$\alpha\mathbf A+\alpha\mathbf B+\beta\mathbf A+\beta\mathbf B=(\alpha+\beta)(\mathbf A+\mathbf B)$$.

A counterexample are any $$\alpha, \beta, \mathbf A$$ and $$\mathbf B$$ s.t. $$\alpha\mathbf B+\beta\mathbf A\ne\mathbf 0$$.