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UPSC 2020 Maths Optional Paper 1 Q1a — Step-by-Step Solution

10 marks · Section A

Subspaces · Linear Algebra · asked 6× in 13 yrs · Read the full method →

Question

Consider the set $V$ of all $n\times n$ real magic squares. Show that $V$ is a vector space over $\mathbb{R}$. Give examples of two distinct $2\times2$ magic squares.

Technique

Subspace test (nonempty + closure) inside $M_n(\mathbb{R})$; line-sum conditions are linear.

Solution

Definition adopted. An $n\times n$ real matrix $A=[a_{ij}]$ is a magic square if there is a common number $s$ (the magic sum, which may depend on $A$) such that every row sum, every column sum, and the two principal diagonal sums all equal $s$:

(We do not require the entries to be distinct positive integers; that combinatorial restriction is not a linear condition. With distinct-integer entries the set is not even closed under addition, so it could never be a vector space. The standard interpretation of this UPSC question is the linear one above.)

Step 1 — $V$ is a subset of the vector space $M_n(\mathbb{R})$

$M_n(\mathbb{R})$, the set of all $n\times n$ real matrices with entrywise addition and scalar multiplication, is a real vector space (standard). It suffices to show $V$ is a subspace: a nonempty subset closed under addition and scalar multiplication. Closure gives all eight axioms by inheritance.

Step 2 — $V$ is nonempty (contains $0$)

The zero matrix $O$ has every row, column and diagonal summing to $0$, so $O\in V$ with magic sum $0$. Hence $V\neq\varnothing$.

Step 3 — Closure under addition

Let $A,B\in V$ with magic sums $s_A,s_B$. For any row $i$,

independent of $i$. The identical computation holds for every column and for both diagonals. Thus $A+B$ has all its rows, columns and diagonals equal to the common value $s_A+s_B$, so $A+B\in V$.

Step 4 — Closure under scalar multiplication

Let $A\in V$ with magic sum $s_A$ and $\lambda\in\mathbb{R}$. For any row $i$,

and likewise for columns and diagonals. Hence $\lambda A\in V$ with magic sum $\lambda s_A$.

Step 5 — Conclusion

$V$ is a nonempty subset of $M_n(\mathbb{R})$ closed under addition and scalar multiplication; therefore $V$ is a subspace of $M_n(\mathbb{R})$ and in particular a vector space over $\mathbb{R}$. $\blacksquare$

Answer