Find the Rank [[1,0,0],[0,1,0],[0,0,1]]*p

Problem

rank([[1,0,0],[0,1,0],[0,0,1]]*p)

Solution

1. Identify the matrix and the scalar. The expression involves the 3×3 identity matrix I multiplied by a scalar p

2. Simplify the expression. Multiplying the identity matrix by a scalar p results in a diagonal matrix.

[[1,0,0],[0,1,0],[0,0,1]]*p=[[p,0,0],[0,p,0],[0,0,p]]

3. Determine the rank based on the value of p The rank of a matrix is the number of linearly independent rows or columns.

4. Analyze the case where p≠0 If p is non-zero, there are 3 linearly independent rows, so the rank is 3.

5. Analyze the case where p=0 If p is zero, the matrix becomes the zero matrix, which has a rank of 0.

Final Answer

rank([[1,0,0],[0,1,0],[0,0,1]]*p)={[3,if *p≠0],[0,if *p=0])

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