Q.62 If 𝑣⃗ = (2, 2, 2) is an eigenvector of the matrix
( 1 2 3 ; 1 2 3 ; 1 2 3 ) corresponding to the non-zero eigenvalue λ,
then the value of λ is ______.

The Matrix and Given Eigenvector

The matrix is:

A =
( 1 2 3 )
( 1 2 3 )
( 1 2 3 )

and the eigenvector is:

𝑣⃗= (2, 2, 2)

We find λ such that:

A 𝑣⃗= λ 𝑣⃗

Step-by-Step Solution

Compute A 𝑣⃗:

• First row: 1×2 + 2×2 + 3×2 = 12
• Second row: 1×2 + 2×2 + 3×2 = 12
• Third row: 1×2 + 2×2 + 3×2 = 12

Thus:

A 𝑣⃗= (12, 12, 12)

Factor out 6:

(12,12,12) = 6(2,2,2) = 6 𝑣⃗

Therefore, the eigenvalue is:

λ = 6

Verification

𝑣⃗≠ (0,0,0), and λ = 6 is non-zero as required.

Why Direct Computation Works

The eigenvector definition:

A 𝑣⃗= λ 𝑣⃗

or equivalently:

(A − λI)𝑣⃗= 0

Since the matrix rows are identical, vectors like (1,1,1) or (2,2,2) are natural eigenvectors.

Characteristic Polynomial (Optional Verification)

Compute:

det(A − λI) = 0

Produces eigenvalues:

• λ = 6 (multiplicity 1)
• λ = 0 (multiplicity 2)

The only non-zero eigenvalue is:

λ = 6

Exam Tips for Eigenvalue Problems

• Given eigenvector? Multiply A 𝑣⃗ immediately — fastest path.
• Identical rows/columns → eigenvector (1,1,…,1).
• For fill-in-the-blank style questions, λ = 6 is exact.