Gate Questions in Hindi- GATE 2025 Symmetric relation The number of different n- multiply -n symmetric matrix.<\/p>\n
[fvplayer id=”231″]<\/p>\n
GATE 2025 \u0915\u0947 \u0932\u093f\u090f \u0939\u093f\u0902\u0926\u0940 \u092e\u0947\u0902 \u092a\u094d\u0930\u0936\u094d\u0928: \u0938\u092e\u092e\u093f\u0924 \u0938\u0902\u092c\u0902\u0927<\/strong><\/p>\n \u092a\u094d\u0930\u0936\u094d\u0928:<\/strong> \u0915\u093f\u0938\u0940 n \u00d7 n \u0938\u092e\u092e\u093f\u0924 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0915\u0940 \u0915\u0941\u0932 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u093f\u0924\u0928\u0940 \u0939\u094b\u0924\u0940 \u0939\u0948?<\/p>\n \u0938\u092e\u093e\u0927\u093e\u0928:<\/strong><\/p>\n \u0938\u092e\u092e\u093f\u0924 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 (Symmetric Matrix) \u0935\u0939 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0939\u094b\u0924\u0940 \u0939\u0948 \u091c\u093f\u0938\u092e\u0947\u0902 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0915\u093e \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0924\u0924\u094d\u0935 \u0905\u092a\u0928\u0947 \u092e\u0941\u0916\u094d\u092f \u0935\u093f\u0915\u0930\u094d\u0923 (main diagonal) \u0915\u0947 \u0938\u093e\u092a\u0947\u0915\u094d\u0937 \u0938\u092e\u092e\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0905\u0930\u094d\u0925\u093e\u0924, \u0915\u093f\u0938\u0940 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 A \u0915\u0947 \u0932\u093f\u090f, \u092f\u0926\u093f A = A\u1d40 (A transpose) \u0939\u094b, \u0924\u094b \u0935\u0939 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0938\u092e\u092e\u093f\u0924 \u0915\u0939\u0932\u093e\u0924\u0940 \u0939\u0948\u0964<\/p>\n n \u00d7 n \u0938\u092e\u092e\u093f\u0924 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u092e\u0947\u0902, \u092e\u0941\u0916\u094d\u092f \u0935\u093f\u0915\u0930\u094d\u0923 (main diagonal) \u0915\u0947 \u0924\u0924\u094d\u0935 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0930\u0942\u092a \u0938\u0947 \u091a\u0941\u0928\u0947 \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u092e\u0941\u0916\u094d\u092f \u0935\u093f\u0915\u0930\u094d\u0923 \u0915\u0947 \u090a\u092a\u0930 \u092f\u093e \u0928\u0940\u091a\u0947 \u0915\u0947 \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u093e \u091a\u092f\u0928 \u0915\u0930\u0928\u0947 \u0915\u0947 \u092c\u093e\u0926, \u0909\u0928\u0915\u0947 \u0938\u092e\u092e\u093f\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u092a\u0930 \u0935\u0939\u0940 \u092e\u093e\u0928 \u0930\u0916\u0947 \u091c\u093e\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f:<\/strong><\/p>\n 3 \u00d7 3 \u0938\u092e\u092e\u093f\u0924 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u092e\u0947\u0902,<\/p>\n \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u0915\u0941\u0932 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e = \u092e\u0941\u0916\u094d\u092f \u0935\u093f\u0915\u0930\u094d\u0923 \u0915\u0947 \u0924\u0924\u094d\u0935 + \u092e\u0941\u0916\u094d\u092f \u0935\u093f\u0915\u0930\u094d\u0923 \u0915\u0947 \u090a\u092a\u0930 \u0915\u0947 \u0924\u0924\u094d\u0935<\/p>\n n \u00d7 n \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u092e\u0947\u0902:<\/p>\n \u0905\u0924\u0903, \u0915\u0941\u0932 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e = n + n(n – 1)\/2 = (n\u00b2 + n)\/2<\/p>\n \u092f\u0926\u093f \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0924\u0924\u094d\u0935 \u0915\u094b k \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u092e\u093e\u0928 \u0926\u093f\u090f \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0915\u0941\u0932 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0938\u092e\u092e\u093f\u0924 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0939\u094b\u0917\u0940:<\/p>\n k^[(n\u00b2 + n)\/2]<\/p>\n \u0909\u0926\u093e\u0939\u0930\u0923:<\/strong><\/p>\n \u092f\u0926\u093f n = 3 \u0914\u0930 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0924\u0924\u094d\u0935 2 \u092e\u093e\u0928 (0 \u092f\u093e 1) \u0932\u0947 \u0938\u0915\u0924\u093e \u0939\u0948, \u0924\u094b<\/p>\n \u0915\u0941\u0932 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e = (3\u00b2 + 3)\/2 = 6<\/p>\n \u0905\u0924\u0903, \u0915\u0941\u0932 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0938\u092e\u092e\u093f\u0924 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e = 2\u2076 = 64<\/p>\n \u0928\u094b\u091f:<\/strong> GATE \u092a\u0930\u0940\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u092a\u094d\u0930\u0936\u094d\u0928 \u092a\u0942\u091b\u0947 \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u091c\u0939\u093e\u0901 \u0938\u092e\u092e\u093f\u0924 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0915\u0940 \u0915\u0941\u0932 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0938\u0902\u0916\u094d\u092f\u093e \u091c\u094d\u091e\u093e\u0924 \u0915\u0930\u0928\u0940 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964<\/p>\n \u0905\u0927\u093f\u0915 \u091c\u093e\u0928\u0915\u093e\u0930\u0940 \u0915\u0947 \u0932\u093f\u090f:<\/strong> \u0906\u092a GATE \u0915\u0940 \u0906\u0927\u093f\u0915\u093e\u0930\u093f\u0915 \u0935\u0947\u092c\u0938\u093e\u0907\u091f \u092a\u0930 \u092a\u093f\u091b\u0932\u0947 \u0935\u0930\u094d\u0937\u094b\u0902 \u0915\u0947 \u092a\u094d\u0930\u0936\u094d\u0928\u092a\u0924\u094d\u0930 \u0926\u0947\u0916 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n In GATE 2025, a common question in Discrete Mathematics or Linear Algebra involves counting the number of symmetric n\u00d7nn \\times n<\/span>n<\/span>\u00d7<\/span><\/span>n<\/span><\/span><\/span><\/span> matrices<\/strong> over a given set (usually binary or integers modulo some number).<\/p>\n Let\u2019s solve:<\/p>\n GATE 2025 \u092e\u0947\u0902 \u090f\u0915 \u092a\u094d\u0930\u0936\u094d\u0928 \u092a\u0942\u091b\u093e \u0917\u092f\u093e \u0939\u0948:<\/strong><\/p>\n “Ek n\u00d7nn \\times n<\/span>n<\/span>\u00d7<\/span><\/span>n<\/span><\/span><\/span><\/span> symmetric matrix kitne alag-alag tarike se banaye ja sakte hain, agar har element kisi finite set se liya jata hai?”<\/p>\n<\/blockquote>\n Let\u2019s assume that each element is from a set with kk<\/span>k<\/span><\/span><\/span><\/span> elements (like {0,1} \u2192 then k=2k=2<\/span>k<\/span>=<\/span><\/span>2<\/span><\/span><\/span><\/span>).<\/p>\n A symmetric matrix AA<\/span>A<\/span><\/span><\/span><\/span> satisfies:<\/p>\n A=ATA = A^T<\/span>A<\/span>=<\/span><\/span>A<\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n Which means:<\/p>\n Diagonal elements can be anything \u2192 There are nn<\/span>n<\/span><\/span><\/span><\/span> diagonal elements.<\/p>\n<\/li>\n Off-diagonal elements are mirrored<\/strong> \u2192 aij=ajia_{ij} = a_{ji}<\/span>a<\/span>ij<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>a<\/span>ji<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n There are n(n\u22121)2\\frac{n(n-1)}{2}<\/span>2<\/span><\/span>n<\/span>(<\/span>n<\/span>\u2212<\/span>1)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> such pairs above the diagonal. So:<\/p>\n Total\u00a0symmetric\u00a0matrices=kn\u00d7kn(n\u22121)2=kn(n+1)2\\text{Total symmetric matrices} = k^{n} \\times k^{\\frac{n(n-1)}{2}} = k^{\\frac{n(n+1)}{2}}<\/span>Total\u00a0symmetric\u00a0matrices<\/span><\/span>=<\/span><\/span>k<\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>k<\/span>2<\/span>n<\/span>(<\/span>n<\/span>\u2212<\/span>1)<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>k<\/span>2<\/span>n<\/span>(<\/span>n<\/span>+<\/span>1)<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n For n=3n = 3<\/span>n<\/span>=<\/span><\/span>3<\/span><\/span><\/span><\/span>, k=2k = 2<\/span>k<\/span>=<\/span><\/span>2<\/span><\/span><\/span><\/span> (binary matrix):<\/p>\n<\/li>\n<\/ul>\n Total=23(3+1)2=26=64\\text{Total} = 2^{\\frac{3(3+1)}{2}} = 2^6 = 64<\/span>Total<\/span><\/span>=<\/span><\/span>22<\/span>3(<\/span>3+<\/span>1)<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>26<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\n
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Question (in Hindi style)<\/strong>:<\/h3>\n
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\nSymmetric Matrix Properties<\/strong>:<\/h3>\n
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\nTotal Number of Symmetric Matrices<\/strong>:<\/h3>\n
\nExample<\/strong>:<\/h3>\n
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