Exercise Prove that

• If A is any m×n matrix,     then AIn=A=ImA
  
• AIn=InA=A for every n×n matrix A.
  

If A=(aij)m×n is anymatrix, the transpose of A isdefined as A′(or AT) =(aji)n×m. The subscripts i and j areinterchanged because every row of A becomesa column of A′, and everycolumn of A becomes a row of A′. A squarematrix is said to be symmetric if A=A′.

The following rules apply to matrixtransposition:

• (A′)′=A
  
• (A+B)′=A′+B′
  
• (αA)′=αA′
  
• (AB)′=B′A′
  

Exercise Show 4 above.

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