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
1 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.