Find the Rank [[1,0,1],[0,1,-1],[-1,0,1]]

Problem

Rank*[[1,0,1],[0,1,−1],[−1,0,1]]

Solution

1. Identify the matrix A and prepare to perform elementary row operations to find its row echelon form.

A=[[1,0,1],[0,1,−1],[−1,0,1]]

2. Eliminate the first element of the third row by adding the first row to the third row ((R_3)→(R_3)+(R_1).

[[1,0,1],[0,1,−1],[0,0,2]]

3. Observe that the matrix is now in row echelon form, where all rows have a leading non-zero entry (pivot).

Pivot 1: *(a_11)=1

Pivot 2: *(a_22)=1

Pivot 3: *(a_33)=2

4. Determine the rank by counting the number of non-zero rows in the row echelon form.

Number of non-zero rows=3

Final Answer

Rank*[[1,0,1],[0,1,−1],[−1,0,1]]=3

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