Find the Rank [[i+1,2,1,-i],[1,i,2i,2]]

Problem

Rank*[[i+1,2,1,−i],[1,i,2*i,2]]

Solution

1. Write the matrix A and identify its dimensions.

A=[[i+1,2,1,−i],[1,i,2*i,2]]

The matrix has 2 rows and 4 columns, so the maximum possible rank is 2.

2. Swap the rows to place a simpler entry in the first row, first column position.

(R_1)↔(R_2)⇒[[1,i,2*i,2],[i+1,2,1,−i]]

3. Eliminate the first entry of the second row by performing the row operation (R_2)→(R_2)−(i+1)*(R_1)

(R_2,1)=(i+1)−(i+1)*(1)=0

(R_2,2)=2−(i+1)*(i)=2−(i2+i)=2−(−1+i)=3−i

(R_2,3)=1−(i+1)*(2*i)=1−(2*i2+2*i)=1−(−2+2*i)=3−2*i

(R_2,4)=−i−(i+1)*(2)=−i−(2*i+2)=−2−3*i

4. Analyze the resulting matrix in row-echelon form.

[[1,i,2*i,2],[0,3−i,3−2*i,−2−3*i]]

The second row is not a row of zeros because 3−i≠0

5. Count the non-zero rows to determine the rank.
   There are 2 linearly independent rows.

Final Answer

Rank*[[i+1,2,1,−i],[1,i,2*i,2]]=2

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