Q.No. 23 The largest eigenvalue of the matrix
[ 4  1
-2   1 ]
is __________.

The largest eigenvalue of the matrix [4 1; -2 1] is 5.

Step-by-Step Solution

To find eigenvalues, solve the characteristic equation det(A − λI) = 0, where

A = [4 1; −2 1].

Form A − λI

[4 − λ   1;
−2      1 − λ]

Determinant

(4 − λ)(1 − λ) − (1)(−2) = λ² − 5λ + 6 = 0

Eigenvalues

Roots are λ = (5 ± √(25 − 24)) / 2 = 5, 1

Largest eigenvalue = 5

Verification Methods

• Trace(A) = 4 + 1 = 5 = sum of eigenvalues (5 + 1)
• det(A) = (4 × 1) − (1 × −2) = 6 = product (5 × 1)
• Power method converges to 5 because |5| > |1|

Characteristic Equation Derivation

Compute:

det[4 − λ   1;
−2      1 − λ] = (4 − λ)(1 − λ) + 2 = λ² − 5λ + 6 = 0

Quadratic formula gives λ = 5, 1.
Largest eigenvalue = 5

Why Largest Eigenvalue Matters

In power method iterations or stability analysis, the dominant eigenvalue governs convergence.

Trace = 5 and det = 6 confirm validity.

Exam Tips for Eigenvalue Problems

• Verify with trace (sum) and determinant (product).
• For 2×2, use characteristic equation λ² − (trace)λ + det = 0.