Consider the 2 andtimes; 2 matrixWhat is the maximum value of x, for which both the Eigen values of matrix are real and positive?Correct answer is between '2.6,2.7'. Can you explain this answer?

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Consider the 2 × 2 matrix

What is the maximum value of x, for which both the Eigen values of matrix are real and positive?

Correct answer is between '2.6,2.7'. Can you explain this answer?

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Consider the 2 × 2 matrixWhat is the maximum value of x, for whi...

⇒ (2 – λ) (4 – λ) – 3x = 0
⇒ λ² – 6λ +8 – 3x = 0

Roots of the above equation are Eigen values and let the Eigen values be λ₁ and λ₂
From the properties of Eigen values,

For the Eigen values to be positive, λ₁λ₂> 0
⇒ 8 – 3x > 0
⇒ x < 8/3
For roots to be real,

⇒ 36 – 32 + 12x > 0
⇒ x > -1/3
The range of for which Eigen values of matrix are real and positive is:

Maximum value of x = 8/3 = 2.67

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