If A is idemponent matrix, then find the value of (A+I)n, ∀n∈N Where I is the identity matrix having the same order of A. - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

A
$I + (2^{n} - 1) A$
B
$I - (2^{n} - 1) A$
C
$3 I + (2^{n} - 1) A$
D
$4 I + (2^{n} - 1) A$

Solution:

$∴ A$ is idemponent matrix

$∴ A^{2} = A$ ,

similarly $A = A^{2} = A^{3} = A^{4} = . . . = A^{n} . . . (i)$

Now, ${(A + I)}^{n} = {(I + A)}^{n}$

$= I + {}^{n}C_{1} A + {}^{n}C_{2} A^{2} + {}^{n}C_{3} A^{3} + . . . + {}^{n}C_{n} A^{n} = I + ({}^{n}C_{1} + {}^{n}C_{2} + {}^{n}C_{3} + . . . + {}^{n}C_{n}) A [f r o m E q . (i)] = I + (2^{n} - 1) A$

Hence, ${(A + I)}^{n} = I + (2^{n} - 1) A, \forall n \in N$ .

If $A$ is idemponent matrix, then find the value of ${(A + I)}^{n}$ $, \forall n \in N$ Where $I$ is the identity matrix having the same order of $A$ .

Solution:

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