2x2 matrice such DA=0

Question

A)find non-zero 2x2 matirx A and D that DA=0

B)show that if A is invertible, and DA=0, then D=0

Answer 1

Let's do part B first:

 

If A is invertible, then A^-1 exists.  

 

So, if DA = 0, then DAA^-1 = 0A^-1 (0 is the 2x2 zero matrix)

 

                            DI₂ = 0  (I₂ is the 2x2 identity matrix)

 

                              D = 0

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A. If D and A are both nonzero 2x2 matrices such that DA = 0, then from part B, neither one of them can be invertible.  

 

If 2 rows of a square matrix are multiples of each other, then the matrix is not invertible because if two rows are multiples of each other, the determinant of the matrix is zero.  

 

So, we need to find two 2x2 nonzero noninvertible matrices D and A so that DA = 0.

            ⌈ 1      2 ⌉                ⌈ a     b ⌉

Let D =                   and A =  

            ⌊ 2      4 ⌋                ⌊ c     d ⌋

 

Since DA = 0, we get a+2c = 0 and b + 2d = 0

 

For example, we could choose a = 2, c = -1 and b = -2, d = 1

 

               ⌈ 1     2 ⌉             ⌈ 2      -2 ⌉

So, if D =                and A =                                                                                                                             ⌊2      4 ⌋             ⌊ -1      1 ⌋                                        

 

      then D and A are nonzero 2x2 matrices where DA = 0.