Find the Rank T[[a,b],[b,c]]

Problem

Rank*[[a,b],[b,c]]

Solution

1. Identify the matrix A as a 2×2 symmetric matrix.

A=[[a,b],[b,c]]

2. Calculate the determinant of the matrix to determine if the rank is full.

det(A)=a*c−b2

3. Determine the rank based on the value of the determinant. If det(A)≠0 the rank is 2.

Rank(A)=2⇔a*c−b2≠0

4. Analyze the case where the determinant is zero. If a*c−b2=0 the rank is 1 unless all entries are zero.

Rank(A)=1⇔a*c−b2=0* and *(a≠0* or *b≠0* or *c≠0)

5. Consider the trivial case where all entries are zero.

Rank(A)=0⇔a=b=c=0

Final Answer

Rank*[[a,b],[b,c]]={[2,if *a*c−b2≠0],[1,if *a*c−b2=0* and *a2+b2+c2≠0],[0,if *a=b=c=0])

Want more problems? Check here!