Properties of Matrix Operations

==2024-12-03

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You are good, so are your properties :LiSmile:.

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Theory

1. $A+B=B+A$
2. $(A+B)+C=A+(B+C)$
3. $A+0=A$
4. $r(A+B)=rA+rB$
5. $A(BC)=(AB)C$
6. $A(B+C)=AB+AC$
7. $(B+C)A=BA+CA$
8. $r(AB)=(rA)B=A(rB)$
9. $I_{m}A=A{I}_n$= $A_{m\times n}$
10. $(A^T{)}^T=A$
11. $(I_n{)}^T=I_n$
12. $A{A}^{-1}=A^{-1}A=I_n$
13. $(AB{)}^{-1}=B^{-1}{A}^{-1}$
14. $(rA{)}^{-1}=B^{-1}{A}^{-1}$ iff $B=rI$ or B is a diagonal matrix
15. $(I_n{)}^{-1}=I_n$
16. $(A^T{)}^{-1}=(A^{-1}{)}^T$
17. $(A^{-1}{)}^{-1}=A$
18. $(\bar{A}{)}^T=\overline{(A^T)}$
19. $\overline{\stackrel{ˉ}{A}}=A$

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Examples

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Implementation

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Q&A

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PTR