Q.39 A 2×2 matrix P has an eigenvalue λ₁ = 2 with eigenvector
x₁ = ( 1 0 )^T
and another eigenvalue λ₂ = 5, with eigenvector
x₂ = ( 1 1 )^T.
The matrix P is

(A)	( 2   0 ) ( 0   5 )
(B)	( 2   3 ) ( 0   5 )
(C)	( 1   1 ) ( 0   1 )
(D)	( 1   1 ) ( 1   0 )

A 2×2 matrix P has:

• Eigenvalue λ₁ = 2 with eigenvector x₁ = (1, 0)
• Eigenvalue λ₂ = 5 with eigenvector x₂ = (1, 1)

We must determine the correct matrix among the options.

Key Concept

For any matrix P, an eigenpair satisfies:

P x = λx

Step 1 — Use First Eigenpair

P(1,0)^T gives the first column of P:

= 2(1,0)^T = (2,0)^T

So first column is (2,0).

Step 2 — Use Second Eigenpair

P(1,1)^T = (first column + second column) = 5(1,1)^T = (5,5)^T

We already know first column = (2,0):

(2,0) + second column = (5,5)

Therefore second column = (3,5)

Final Matrix

P =

(2 3)

(0 5)

Correct Answer

Option B

Option-Wise Explanation

(B) (2 3; 0 5) — Correct

Matches both eigenpairs:

• Column one gives λ₁ = 2
• Sum of columns gives λ₂ = 5

(A) (2 0; 0 5) — Incorrect

Diagonal matrix has eigenvectors (1,0) and (0,1), not (1,1).

(C) (1 1; 0 1) — Incorrect

Both eigenvalues are 1; cannot match λ₁ = 2 and λ₂ = 5.

(D) (1 1; 1 0) — Incorrect

Eigenvalues are (1 ± √5) / 2, not 2 and 5.