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Consider the following statements P and Q:
(P) : IfM =   then M is singular.
(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?
Verified Answer
Consider the following statements P and Q:(P) : IfM =then M is singula...
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 -λ1 ­will 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)
(c) If λ is an eigen value of matrix A, then  will be an eigen value of matrix  where K is a scalar.
From the above properties, eigen values of T are,

As λ1, λ2, λ3 are distinct values, λ′1,λ′2,λ′3 will be distinct.
Hence, matrix T is diagonalizable
So, only Q is true.
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Consider the following statements P and Q:(P) : IfM =then M is singular.(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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