Consider the following statements P and Q:(Q): Let S be a diagonalizable matrix. If T is a matrix such that S + 5 T = I, then T is diagonalizable.Which of the above statements hold TRUE?a)Both P and Qb)Only Pc)Only Qd)Neither P nor QCorrect answer is option 'C'. Can you explain this answer?

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Consider the following statements P and Q:

(Q): Let S be a diagonalizable matrix. If T is a matrix such that S + 5 T = I, then T is diagonalizable.

Which of the above statements hold TRUE?

a)
Both P and Q
b)
Only P
c)
Only Q
d)
Neither P nor Q

Correct answer is option 'C'. Can you explain this answer?

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Consider the following statements P and Q:(Q): Let S be a diagonalizab...

A matrix is said to be singular, if determinant of that matrix is zero.

= 1 (18 – 12) - 1 (9 – 4) + 1 (3 – 2)

= 6 – 5 + 1 = 2 ≠ 0

M is non singular

(Q) A matrix can be diagonalizable when it has distinct eigen values

S is a diagonalizable matrix. Hence, has distinct eigen values.

Let S be a 3 × 3 matrix and the eigen values of s are λ₁, λ₂, λ₃

Given that, S + 5T = I

From the properties of Eigen values,

(a) If λ₁ is an eigen value of matrix A, then -λ₁will be on eigen value of matrix -A.

(b) If λ₁ is an eigen value of matrix A, then (λ₁ + 1) will be an eigen value of matrix (A + I)

From the above properties, eigen values of T are,

As λ₁, λ₂, λ₃ are distinct values,

will be distinct.

Hence, matrix T is diagonalizable

So, only Q is true.

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Consider the following statements P and Q:(Q): Let S be a diagonalizable matrix. If T is a matrix such that S + 5 T = I, then T is diagonalizable.Which of the above statements hold TRUE?a)Both P and Qb)Only Pc)Only Qd)Neither P nor QCorrect answer is option 'C'. Can you explain this answer?

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