Ex.1.1 Q7 Integers - NCERT Maths Class 7

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Question

In a magic square, each row, column and the diagonal has the same sum. Check which of the following is a magic square?

 $$5$$ $$–1$$ $$–4$$ $$–5$$ $$–2$$ $$–7$$ $$0$$ $$3$$ $$–3$$
 1 $$–10$$ $$0$$ $$–4$$ $$–3$$ $$–2$$ $$–6$$ $$4$$ $$–7$$

Text Solution

What is known?

Value of each row, column and the diagonal.

What is unknown?

To check which row, columnand the diagonal have same sum.

Reasoning:

Use the table and find the sum of the row, column and the diagonal.

Steps:

In square (i)

(a) Taking rows

\begin{align} {R_1} &= 5 + \left( {-1} \right) + \left( {-4} \right) = 0\\{R_2}& = -5 + \left( {-2} \right) + 7 = 0\\{R_3} &= \,\,\,0 + 3 + \left( {-3} \right) = 0\end{align}

(b) Taking columns

\begin{align}{C_1} &= {\rm{5 + }}\left( {{\rm{-5}}} \right){\rm{ + 0}}\,{\rm{ = }}\,{\rm{0}}\\ {C_2} &=\rm{-1 + }\left( \rm{-2} \right){\rm{ + }}\left( {{\rm{-3}}} \right){\rm{ = }}\,{\rm{0}}\\{C_3} &= \rm{-}\,{\rm{4 + 7 + }}\left( {{\rm{-3}}} \right){\rm{ = }}\,\,{\rm{0}}\end{align}

(c) Taking diagonals

\begin{align}{d_1} &= 5 + \left( {-2} \right) + \left( {-3} \right) = 0\\{d_2} &= -\,4 + \left( {-2} \right) + 0 = -\,6\end{align}

This square is not a magic square because the sum of one of its diagonal is not equal to the sum of its other diagonal,

In square (ii)

(a) Taking rows

\begin{align}{R_1} &= 1 + \left( {-10} \right) + 0 = -9\\{R_2} &= -4 + \left( {-3} \right) + \left( {-2} \right) = -9\\{R_3} &= \,-6 + 4 + \left( {-7} \right) = -9\end{align}

(b) Taking columns

\begin{align}{C_1} &= 1 + \left( {-4} \right) + \left( {-6} \right) = -9\\{C_2} &= -10 + \left( {-3} \right) + 4 = -9\\{C_3} &= 0 + \left( {-2} \right) + \left( {-7} \right) = -9\end{align}

(c) Taking diagonals

\begin{align}{d_1} &= 1 + \left( {-3} \right) + \left( {-7} \right) = -9\\{d_2} &= 0 + \left( {-3} \right) + \left( {-6} \right) = -9\end{align}

This square box is a magic square because the sum of its rows, columns and diagoanls are equal.

Hence, (ii) is a magic square.

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