JEE MAIN - Mathematics (2021 - 22th July Evening Shift - No. 8)
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 :
2
1
3
9
Explanation
$$A = \left[ {\matrix{
{{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr
{{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr
{{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr
} } \right]$$
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 $$x = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right]$$
$$AX = \left[ {\matrix{ {{a_{11}} + {a_{12}} + {a_{13}}} \cr {{a_{21}} + {a_{22}} + {a_{23}}} \cr {{a_{31}} + {a_{32}} + {a_{33}}} \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right]$$
$$\Rightarrow$$ 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
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