If A and P are different matrices of order n satisfying A3=P3 and A2 P=P2 A (where .|A-P| ≠ 0) then mid A2+P2 is equal to

Question

If A and P are different matrices of order n satisfying A3=P3 and A2 P=P2 A (where .|A-P| ≠ 0) then mid A2+P2 is equal to (A) n (B) 0 (C) |A | P| (D) |A+P|

Tardigrade Admin · Accepted Answer

(A2+P2)(A-P) =A3-A2 P+P2 A-P3 =(A3-P3)+(P2 A-A2 P) =0 ∴ mid(A2+P2)(A-P) mid=0 ∴ |A2+P2|=0 (∵|A-P| ≠ 0)

Q. If $A$ and $P$ are different matrices of order $n$ satisfying $A^{3} = P^{3}$ and $A^{2} P = P^{2} A$ (where $∣ A - P ∣ \neq = 0)$ then $∣ A^{2} + P^{2}$ is equal to

$n$

0

$∣ A ∥ P ∣$

$∣ A + P ∣$

$(A^{2} + P^{2}) (A - P) = A^{3} - A^{2} P + P^{2} A - P^{3}$
$= (A^{3} - P^{3}) + (P^{2} A - A^{2} P)$
$= 0$
$∴ ∣ (A^{2} + P^{2}) (A - P) ∣= 0$
$∴ ∣ ∣ A^{2} + P^{2} ∣ ∣ = 0 (∵ ∣ A - P ∣ \neq = 0)$

Q. If A and P are different matrices of order n satisfying A3=P3 and A2P=P2A (where ∣A−P∣=0) then ∣A2+P2 is equal to

n

0

∣A∥P∣

∣A+P∣

(A2+P2)(A−P)=A3−A2P+P2A−P3 =(A3−P3)+(P2A−A2P) =0 ∴∣(A2+P2)(A−P)∣=0 ∴∣∣​A2+P2∣∣​=0(∵∣A−P∣=0)

Q. If $A$ and $P$ are different matrices of order $n$ satisfying $A^{3} = P^{3}$ and $A^{2} P = P^{2} A$ (where $∣ A - P ∣ \neq = 0)$ then $∣ A^{2} + P^{2}$ is equal to

$n$

$∣ A ∥ P ∣$

$∣ A + P ∣$

$(A^{2} + P^{2}) (A - P) = A^{3} - A^{2} P + P^{2} A - P^{3}$
$= (A^{3} - P^{3}) + (P^{2} A - A^{2} P)$
$= 0$
$∴ ∣ (A^{2} + P^{2}) (A - P) ∣= 0$
$∴ ∣ ∣ A^{2} + P^{2} ∣ ∣ = 0 (∵ ∣ A - P ∣ \neq = 0)$