"Let A = [aij] be a real matrix of order 3 $\times$ 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A3 is equal to :"

Question

Accepted Answer

"$$A = \left[ {\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right]$$

Let $x = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right]$

$$AX = \left[ {\matrix{ {{a_{11}} + {a_{12}} + {a_{13}}} \cr {{a_{21}} + {a_{22}} + {a_{23}}} \cr {{a_{31}} + {a_{32}} + {a_{33}}} \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right]$$

$\Rightarrow$ AX = X

Replace X by AX

A²X = AX = X

Replace X by AX

A³X = AX = X

Let $${A^3} = \left[ {\matrix{ {{x_1}} & {{x_2}} & {{x_3}} \cr {{y_1}} & {{y_2}} & {{y_3}} \cr {{z_1}} & {{z_2}} & {{z_3}} \cr } } \right]$$

$${A^3}\left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right] = \left[ {\matrix{ {{x_1}} & {{x_2}} & {{x_3}} \cr {{y_1}} & {{y_2}} & {{y_3}} \cr {{z_1}} & {{z_2}} & {{z_3}} \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right]$$

Sum of all the element = 3"

Let A = [a_ij] be a real matrix of order 3 $\times$ 3, such that a_i1 + a_i2 + a_i3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A³ is equal to :

Solution

About this question

Drill 25 more like these. Every day. Free.

Let A = [aij] be a real matrix of order 3 $\times$ 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A3 is equal to :

Solution

About this question

More Matrices and Determinants questions

Drill 25 more like these. Every day. Free.

Let A = [a_ij] be a real matrix of order 3 $\times$ 3, such that a_i1 + a_i2 + a_i3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A³ is equal to :