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. \(\alpha\mathbf B+\beta\mathbf A\ne\mathbf 0\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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