A square matrix A of order 3 satisfies A2=Ι-2A , where Ι is an identity matrix of order 3 . If An=29A-12Ι , then the value of n is equal to

Question

A square matrix A of order 3 satisfies A2=Ι-2A , where Ι is an identity matrix of order 3 . If An=29A-12Ι , then the value of n is equal to (A) 3 (B) 4 (C) 5 (D) 6

Tardigrade Admin · Accepted Answer

A3=A2A=(Ι - 2 A)A=A-2A2=A-2(Ι - 2 A)=5A-2Ι A4=A3A=(5 A - 2 Ι)A=5A2-2A=5(Ι - 2 A)-2A=5Ι-12A A5=A4(A)=(5 Ι - 12 A)A=5A-12A2=5A-12(Ι - 2 A)=29A-12Ι

Q. A square matrix $A$ of order $3$ satisfies $A^{2} = I - 2 A$ , where $I$ is an identity matrix of order $3$ . If $A^{n} = 29 A - 12I$ , then the value of $n$ is equal to

$3$

$4$

$5$

$6$

$A^{3} = A^{2} A = (I - 2 A) A = A - 2 A^{2} = A - 2 (I - 2 A) = 5 A - 2I$
$A^{4} = A^{3} A = (5 A - 2I) A = 5 A^{2} - 2 A = 5 (I - 2 A) - 2 A = 5I - 12 A$
$A^{5} = A^{4} (A) = (5I - 12 A) A = 5 A - 12 A^{2} = 5 A - 12 (I - 2 A) = 29 A - 12I$

Q. A square matrix A of order 3 satisfies A2=I−2A , where I is an identity matrix of order 3 . If An=29A−12I , then the value of n is equal to

3

4

5

6

A3=A2A=(I−2A)A=A−2A2=A−2(I−2A)=5A−2I A4=A3A=(5A−2I)A=5A2−2A=5(I−2A)−2A=5I−12A A5=A4(A)=(5I−12A)A=5A−12A2=5A−12(I−2A)=29A−12I

Q. A square matrix $A$ of order $3$ satisfies $A^{2} = I - 2 A$ , where $I$ is an identity matrix of order $3$ . If $A^{n} = 29 A - 12I$ , then the value of $n$ is equal to

$3$

$4$

$5$

$6$

$A^{3} = A^{2} A = (I - 2 A) A = A - 2 A^{2} = A - 2 (I - 2 A) = 5 A - 2I$
$A^{4} = A^{3} A = (5 A - 2I) A = 5 A^{2} - 2 A = 5 (I - 2 A) - 2 A = 5I - 12 A$
$A^{5} = A^{4} (A) = (5I - 12 A) A = 5 A - 12 A^{2} = 5 A - 12 (I - 2 A) = 29 A - 12I$