Let I be an identity matrix of order 2x2 and P = [[2, -1], [5, -3]]. Then the value of n ∈ N for which Pⁿ = 5I - 8P is equal to......

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    Answered by

    Vishal Baghel | Contributor-Level 10

    a month ago

    Given the matrix P = [2, -1], [5, -3].
    The characteristic equation is det (P - λI) = 0, which is (2-λ) (-3-λ) - (-1) (5) = 0.
    This simplifies to λ² + λ - 1 = 0.
    By the Cayley-Hamilton theorem, the matrix P satisfies this equation: P² + P - I = 0, so P² = I - P.
    To find P³: P³ = P * P² = P (I-P) = P - P² = P - (I-P) = 2P - I.
    The problem asks for N=6, likely related to a higher power P? Continuing the pattern:
    P? = 2P² - P = 2 (I-P) - P = 2I - 3P.
    P? = 2P - 3P² = 2P - 3 (I-P) = 5P - 3I.
    P? = 5P² - 3P = 5 (I-P) - 3P = 5I - 8P.
    The solution N=6 must relate to a di

    ...more

Similar Questions for you

A
alok kumar singh

Let   A = [ x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 ]

Given  A = [ 1 0 1 ] = [ 2 0 2 ]   ...(1)


[ x 1 + z 1 x 2 + z 2 x 3 + z 3 ] = [ 2 0 2 ]

∴    x1 + z1 = 2                … (2)

x2 + z2 = 0               … (3)

x3 + z3 = 0                … (4)

Given   A = [ 1 0 1 ] = [ 4 0 4 ]

[ x 1 + z 1 x 2 + z 2 x 3 + z 3 ] = [ 4 0 4 ]

⇒   – x1 + z1 = −4             … (5)

–x2 + z2 = 0             &nbs

...more
V
Vishal Baghel

f ( x ) = a x 2 + b x + c  

g (x) = px + q

  f ( g ( x ) ) = a ( p x + q ) 2 + b ( p ) ( + q ) + c

8 x 2 2 x = a ( p x + q ) 2 + b ( p x + q ) + c

Compare 8 = ap2 …………… (i)

-2 = a (2pq) + bp

0 = aq2 + bq + c

  9 ( f ( x ) ) = p ( a x 2 + b x + c ) + q

=>4x2 + 6x + 1 = apx2 + bpx + cp + q

=> Andhra Pradesh = 4 ……………. (ii)

6 = bp

1 = cp + q

From (i) & (ii), p = 2, q = -1

=> b = 3, c = 1, a = 2

f (x) = 2x2 + 3x + 1

f (2) = 8 + 6 + 1 = 15

g (x) = 2x – 1

g (2) = 3

V
Vishal Baghel

Kindly consider the following figure

B = (I – adjA)5

V
Vishal Baghel

Kindly consider the following figure

B = (I – adjA)5

R
Raj Pandey

System of equation is

( 2 3 1 1 1 1 1 1 | λ | ) ( x y z ) = [ 2 4 4 λ 4 ]

R1 – 2 R2, R3 – R2

( 0 1 3 1 1 1 0 2 | λ | 1 ) ( x y z ) = ( 1 0 4 4 λ 8 )

System of equation will have no solution for = -7.

 

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