A square matrix P satisfies P 2- I -2 P- where I is identify matrix- If P 2- P 3- P 4- a 2 I - b 2 P- thenA- a 2- b 4 is a perfect squareB- a 4- b 2 is a perfect squareC- a2-b2 is prime D- a 4- b 4 is prime

The correct options are B a^4 +b^2 is a perfect square nbsp; C a^2 +b^2 is prime D a^4 +b^4 is primeP^3=(I 2P)P=P 2(I 2P)=5P 2I nbsp; P^4=(5P 2I)P=5(I 2P) 2 ...

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Byju's Answer

Standard XII

Mathematics

Greatest Integer Function

A square matr...

Question

A square matrix $P$ satisfies $P^{2} = I - 2 P$ , where $I$ is identify matrix. If $P^{2} + P^{3} + P^{4} = a^{2} I - b^{2} P$ , then

a^{2} + b^{4}

is a perfect square

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a^{4} + b^{2}

is a perfect square

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a^{2} + b^{2}

is prime

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a^{4} + b^{4}

is prime

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Solution

The correct options are
B $a^{4} + b^{2}$ is a perfect square
C $a^{2} + b^{2}$ is prime
D $a^{4} + b^{4}$ is prime
$P^{3} = (I - 2 P) P = P - 2 (I - 2 P) = 5 P - 2 I$
$P^{4} = (5 P - 2 I) P = 5 (I - 2 P) - 2 P = 5 I - 12 P$
now check for all options

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Similar questions

If X = 1 - a^{2} and Y = 1 - b^{2}, then what is X^{2} + Y^{2} ?

Q. If a,b,c are in continued proportion prove that :
i)

\frac{a^{2} + a b + b^{2}}{b^{2} + b c + c^{2}} = \frac{a}{c}

.
ii)

\frac{a^{4} + a^{2} b^{2} + b^{4}}{b^{4} + b^{2} c^{2} + c^{4}} = \frac{a^{2}}{c^{2}}

Q. If one factor of

a^{4} + a^{2} b^{2} + b^{4}

a^{2} + b^{2} + a b

, then the other factor is

Q. If in a

△ A B C, a^{4} + b^{4} + c^{4} = 2 a^{2} b^{2} + b^{2} c^{2} + 2 c^{2} a^{2},

then

sin A

is equal to

Q. The number of pairs (a, b) of positive real numbers satisfying

a^{4} + b^{4} < 1

and

a^{2} + b^{2} > 1

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A square matrix P satisfies P2=I−2P, where I is identify matrix. If P2+P3+P4=a2I−b2P, then

The correct options are B a4+b2 is a perfect square C a2+b2 is prime D a4+b4 is primeP3=(I−2P)P=P−2(I−2P)=5P−2I P4=(5P−2I)P=5(I−2P)−2P=5I−12P now check for all options

A square matrix $P$ satisfies $P^{2} = I - 2 P$ , where $I$ is identify matrix. If $P^{2} + P^{3} + P^{4} = a^{2} I - b^{2} P$ , then

The correct options are
B $a^{4} + b^{2}$ is a perfect square
C $a^{2} + b^{2}$ is prime
D $a^{4} + b^{4}$ is prime
$P^{3} = (I - 2 P) P = P - 2 (I - 2 P) = 5 P - 2 I$
$P^{4} = (5 P - 2 I) P = 5 (I - 2 P) - 2 P = 5 I - 12 P$
now check for all options