Let A =[ a ij ] be a square matrix of order 3 such that ai j=2j-i, for all i, j=1,2,3. Then, the matrix A 2+ A 3+ ldots+ A 10 is equal to :

Question

Let A =[ a ij ] be a square matrix of order 3 such that ai j=2j-i, for all i, j=1,2,3. Then, the matrix A 2+ A 3+ ldots+ A 10 is equal to : (A) ((310-3/2)) A (B) ((310-1/2)) A (C) ((310+1/2)) A (D) ((310+3/2)) A

Tardigrade Admin · Accepted Answer

A = beginpmatrix 1 2 22 1 / 2 1 2 1 / 22 1 / 2 1 endpmatrix A 2=3 A A 3=32 A A 2+ A 3+ ldots . A 10 =3 A +32 A + ldots+39 A =(3(39-1)/3-1) A =(310-3/2) A

Q. Let $A = [a_{ij}]$ be a square matrix of order $3$ such that $a_{ij} = 2^{j - i}$ , for all i, $j = 1, 2, 3$ . Then, the matrix $A^{2} + A^{3} + \dots + A^{10}$ is equal to :

$(\frac{3 ^{10} - 3}{2}) A$

$(\frac{3 ^{10} - 1}{2}) A$

$(\frac{3 ^{10} + 1}{2}) A$

$(\frac{3 ^{10} + 3}{2}) A$

$A = ⎝ ⎛ 1 1/2 1/ 2^{2} 21 1/2 2^{2} 21 ⎠ ⎞$
$A^{2} = 3 A$
$A^{3} = 3^{2} A$
$A^{2} + A^{3} + \dots . A^{10}$
$= 3 A + 3^{2} A + \dots + 3^{9} A = \frac{3 ( 3 ^{9} - 1 )}{3 - 1} A$
$= \frac{3 ^{10} - 3}{2} A$

Q. Let A=[aij​] be a square matrix of order 3 such that aij​=2j−i, for all i, j=1,2,3. Then, the matrix A2+A3+…+A10 is equal to :

(2310−3​)A

(2310−1​)A

(2310+1​)A

(2310+3​)A