Find the Rank A=[[i,-5,0],[1,i,-1]]

Problem

A=[[i,−5,0],[1,i,−1]]

Solution

1. Identify the dimensions of the matrix A Since A is a 2×3 matrix, the maximum possible rank is the minimum of the number of rows and columns, which is 2

2. Perform row operations to reach row echelon form. Swap the first and second rows to place a 1 in the first pivot position.

(R_1)↔(R_2)

[[1,i,−1],[i,−5,0]]

3. Eliminate the entry below the first pivot by subtracting i times the first row from the second row.

(R_2)→(R_2)−i*(R_1)

[[1,i,−1],[i−i(1),−5−i(i),0−i*(−1)]]

4. Simplify the entries in the second row using the identity i2=−1

−5−i2=−5−(−1)=−4

0−i*(−1)=i

[[1,i,−1],[0,−4,i]]

5. Determine the rank by counting the number of non-zero rows in the row echelon form. There are 2 non-zero rows.

Final Answer

rank(A)=2

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