Question

If x and y are two non zero matrices of the same order, such that xy = 0, then

Answer 1

Given X ≠ 0 and Y ≠ 0 and XY = 0

If ∣ X ∣≠0 and ∣ Y ∣=0 : ⇒ X−1 exist

Now XY = 0  ⇒ X−1(XY)=X−1.0

                    ⇒ (X−1.X).Y=0      (∴ Matrix multiplication is associative)

                     ⇒ I.Y=0

                     ⇒ Y=0 , not possible since Y≠ 0

If ∣ X ∣=0 and ∣ Y ∣≠0 : ⇒ Y−1 exist

                         Now XY = 0  ⇒ (X(Y)X−1)=0.Y−1

                           (X(YY−1))=0

                  ⇒X.I=0⇒X=0, , not possible since X = 0

→ If∣ X ∣≠0 and ∣ Y ∣≠0

Now XY = 0 ⇒ ∣ X Y ∣=0

⇒ ∣ X∣ .∣Y ∣=0

⇒ ∣ X ∣=0 or ∣Y ∣=0 or ∣ X ∣ =0 & ∣Y∣=0

∴  from the above observations ∣X∣=0 &∣Y∣=0

Answer 2

if x is an n by n matrix with elements aij , where i the row and j is column containing aij. Similarly if y is a matrix with elements bij, such that the number of columns of x is equal to the number of rows of y, the their product is a matrix all elements of which are zeros provided

the sums aij bji =0 for all i and j.

x simple example is the following x11=2, x12=1,x21=4, x22=2

y11=-1, y12=-2, y21=2 and y22=4

thus c11.y11+x12.y21= -2+2=0

x11.y12 +x12.y22= -4+4=0

similarly the other two elements.
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BRAINLIEST PLZ