Rules of matrix calculations.
Task number: 2418
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. \(\alpha\mathbf B+\beta\mathbf A\ne\mathbf 0\).
