Home
Class 12
MATHS
Let A, B, C, D be (not necessarily ) r...

Let ` A, B, C, D ` be (not necessarily ) real matrices such that `A^(T) = BCD , B^(T) =CDA, C^(T) = DAB and D^(T) =ABC` for the matrix `S = ABCD` the least value of k such that `S^(k) = S` is

Text Solution

Verified by Experts

The correct Answer is:
3
`S = ABCD = A ( BCD) = A A^(T)` …(i) `therefore S^(3) = (ABCD) (ABCD) (ABCD)`
`= (ABC) (DAB) (CDA) (BCD)`
`= D^(T) C^(T) B^(T) A^(T) = (BCD)^(T) A^(T)`
`= (A^(T))^(T) A^(T) = A A^(T) = S`
`rArr S^(3) = S`
Hence, least valuse of k is 3.
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    ARIHANT MATHS|Exercise Matrices Exercise 5 : (Matching Type Questions )|4 Videos

  • MATRICES

    ARIHANT MATHS|Exercise Exercise (Statement I And Ii Type Questions)|10 Videos

  • MATRICES

    ARIHANT MATHS|Exercise Exercise (Passage Based Questions)|16 Videos

  • MATHEMATICAL INDUCTION

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|2 Videos

  • MONOTONICITY MAXIMA AND MINIMA

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|31 Videos

Similar Questions

Explore conceptually related problems

Determine K_(p) for the reaction : A(g) + B(s) hArr C(g) + D(g), K_(c) = 49 at 127^(@)C .

The equation of an a.c. voltage is V = 200 sin 50 pi t Then, the r.m.s. value of voltage is

Let f(x)=x+2|x+1|+x-1| . If f(x)=k has exactly one real solution, then the value of k is (a) 3 (b) 0 (c) 1 (d) 2

(In T.S. anther, identify a, b and c)

If P is an orthogonal matrix and Q=P A P^T an dx=P^T A b. I c. A^(1000) d. none of these

Let r ,s ,and t be the roots of equation 8x^3+1001 x+2008=0. Then find the value of (r+s)^3 + (s+t)^3 + (t+r)^3 is .

Let A and B be 3xx3 matrtices of real numbers, where A is symmetric, "B" is skew-symmetric , and (A+B)(A-B)=(A-B)(A+B)dot If (A B)^t=(-1)^k A B , where . (A B)^t is the transpose of the mattix A B , then find the possible values of kdot

f and h are function from A rarr B, where A = {a, b, c, d} and B = {s, t, u} defined as follows f(a) = t, f (b) = s, f(c ) = s f(d) = u, h(a) = s, h(b) = t h(c ) = s, h(a) = u, h(d) = u Which one of the following statement is true?