Let T be a linear operator on R3(R), defined by,T(a , b, c) = ( 3a , a - b , 2a+ b + c ) , then (T2 — I )(T — 3I) is,(where I be a linear operator on R3)a)(O,a -b,2a + b- c)b)(0, a - 4b, 2a + b - 2c )c)( 0. 0. 0)d)None of these.Correct answer is option 'C'. Can you explain this answer?

Question

Let T be a linear operator on R3(R), defined by,T(a , b, c) = ( 3a , a - b , 2a+ b + c ) , then (T2 &mdash; I )(T &mdash; 3I) is,(where I be a linear operator on R3)a)(O,a -b,2a + b- c)b)(0, a - 4b, 2a + b - 2c )c)( 0. 0. 0)d)None of these.Correct answer is option 'C'. Can you explain this answer?

Answer

To find T^2, we first need to find the composition of T with itself.

Let's apply T to the vector (a, b, c):

T(a, b, c) = (3a, a - b, 2a + b + c)

Now let's apply T to the vector (3a, a - b, 2a + b + c):

T(3a, a - b, 2a + b + c) = (3(3a), 3a - (a - b), 2(3a) + (a - b) + (2a + b + c))
= (9a, 3a - a + b, 6a + a - b + 2a + b + c)
= (9a, 2a + b, 9a + 2a + c)

Therefore, T^2(a, b, c) = (9a, 2a + b, 9a + 2a + c).

Similar Mathematics Doubts

The linear operator L(x) is defined by the cross product L(x) = b x X, where b = [0 1 0]T and X = [x1 x2 x3]T arethree dimensional vectors. The 3 x 3 matrix M o f this operation satisfies L (x ) =Then, the eigen values of Mare

Let V be a vector space and T is a linear operator on V. If W is a subspace of V, then Wis invariant under Tiff acTimplies A Tα=0 B TaeW C D Ta=a None of these?

Let T be a linear operator on 3. Let f(X) ∈ [X] denote its characteristic polynomial. Consider the following statements.(

Let F be any field and let T be a linear operator on F² defined by T(a, b)=(a b,

Let v be the vector space of 2x2 matrices over R and let. M=[1 2;3 4] Let T be the linear operator on V defined by T(

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams