{"id":1000,"date":"2024-03-22T11:33:54","date_gmt":"2024-03-22T05:33:54","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1000"},"modified":"2024-10-24T18:11:20","modified_gmt":"2024-10-24T12:11:20","slug":"determining-inverse-matrix","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/determining-inverse-matrix\/","title":{"rendered":"\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u09c7\u09b0 \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (Different processes of determining inverse matrix)"},"content":{"rendered":"

\u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 <\/b>\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be\u09b0 \u0989\u09aa\u09be\u09df (Different processes of determining Inverse Matrix)<\/b><\/span><\/h2>\n

\"\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4<\/p>\n

(i)<\/span> \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (Solution method):<\/b><\/span><\/h3>\n

AX=B<\/span> \u09b9\u09b2\u09c7 X=A^{-1}B<\/span><\/span>,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span>A<\/span> \u098f\u0995\u099f\u09bf \u0985\u09ac\u09cd\u09af\u09a4\u09bf\u0995\u09cd\u09b0\u09ae\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8<\/a><\/span>, X=\\begin{bmatrix}\n\nx\\\\\n\ny\\\\\n\nz\n\n\\end{bmatrix}<\/span><\/span>\u00a0\u098f\u09ac\u0982 B=\\begin{bmatrix}\n\na\\\\\n\nb\\\\\n\nc\n\n\\end{bmatrix}<\/span><\/span>\u00a0\u00a0<\/span><\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3<\/b> : A=\\begin{vmatrix}\n\n0 & 1 & 2\\\\\n\n1 & 2 & 3\\\\\n\n3 & 1 & 1\n\n\\end{vmatrix}<\/span><\/span>\u00a0\u09b9\u09b2\u09c7 <\/b>A^{-1}<\/span>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964\u00a0<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>AX=B<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 X=\\begin{bmatrix}\n\nx\\\\\n\ny\\\\\n\nz\n\n\\end{bmatrix}\n\nB=\\begin{bmatrix}\n\na\\\\\n\nb\\\\\n\nc\n\n\\end{bmatrix}\\\\<\/span><\/span>\u00a0\u09a4\u09be\u09b9\u09b2\u09c7 X=A^{-1}B<\/span><\/span><\/p>\n

\u098f\u0996\u09a8,<\/span> AX=B\\Rightarrow \\begin{bmatrix}\n\n0 & 1 & 2\\\\\n\n1 & 2 & 3\\\\\n\n3 & 1 & 1\n\n\\end{bmatrix}\\begin{bmatrix}\n\nx\\\\\n\ny\\\\\n\nz\n\n\\end{bmatrix}=\\begin{bmatrix}\n\na\\\\\n\nb\\\\\n\nc\n\n\\end{bmatrix}<\/span><\/p>\n\\Rightarrow 0x+y+2z=a......(i)\\\\\n\nx+2y+3z=b......(ii)\\\\\n\n3x+y+z=c......(iii)<\/span>\n3\\times (ii)-(iii)\\Rightarrow 5y+8y=3b-c......(iv)\\\\\n\n5\\times (i)-(iv)\\Rightarrow 2z=5a-3b+c\\Rightarrow z\\frac{5}{2}a-\\frac{3}{2}b+\\frac{1}{2}c\\\\\n\n(i)\\Rightarrow y+5a-3b+c=a\\Rightarrow y=-4a+3b-c\\\\\n\n(iii)\\Rightarrow 3x-4a+3b-c+\\frac{5}{2}a-\\frac{3}{2}b+\\frac{1}{2}c=c\\\\\n\n\\Rightarrow 3x=\\Big(4-\\frac{5}{2}\\Big)a+\\Big(-3+\\frac{3}{2}\\Big)b+\\Big(2-\\frac{1}{2}\\Big)c\\Rightarrow 3x=\\frac{3}{2}a-\\frac{3}{2}b+\\frac{3}{2}c\\Rightarrow x=\\frac{1}{2}a-\\frac{1}{2}b+\\frac{1}{2}c<\/span>\n

\\therefore \\begin{bmatrix}\n\nx\\\\\n\ny\\\\\n\nz\n\n\\end{bmatrix}=\\begin{bmatrix}\n\n1\/2 & -1\/2 & 1\/2\\\\\n\n-4 & 3 & -1\\\\\n\n5\/2 & -3\/2 & 1\/2\n\n\\end{bmatrix}\\begin{bmatrix}\n\na\\\\\n\nb\\\\\n\nc\n\n\\end{bmatrix}<\/span> \u0995\u09be\u099c\u09c7\u0987 A^{-1}=\\begin{bmatrix}\n\n1\/2 & -1\/2 & 1\/2\\\\\n\n-4 & 3 & -1\\\\\n\n5\/2 & -3\/2 & 1\/2\n\n\\end{bmatrix}=-\\frac{1}{2}\\begin{bmatrix}\n\n-1 & 1 & -1\\\\\n\n8 & -6 & 2\\\\\n\n-5 & 3 & -2\n\n\\end{bmatrix}<\/span><\/span><\/p>\n

(ii) \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (Adjoint matrix method):<\/b><\/span><\/h3>\n

\u0995\u09cb\u09a8\u09cb \u0985\u09ac\u09cd\u09af\u09a4\u09bf\u0995\u09cd\u09b0\u09ae\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 <\/span>A<\/span> \u098f\u09b0 \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u0995\u09c7 \u09a4\u09be\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995 <\/span>A<\/span> \u09ae\u09be\u09a8 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09c7 \u09af\u09c7 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df \u09a4\u09be\u0995\u09c7 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 <\/span>A\u00a0<\/span>\u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09ac\u09b2\u09be \u09b9\u09df \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0995\u09cb\u09a8\u09cb \u0985\u09ac\u09cd\u09af\u09a4\u09bf\u0995\u09cd\u09b0\u09ae\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 <\/span>A<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995 \u098f\u09b0 \u09b8\u09b9\u0997\u09c1\u09a3\u0995\u09b8\u09ae\u09c2\u09b9 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09a0\u09bf\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u09c7\u09b0 \u0995\u09b2\u09be\u09ae\u09c7\u09b0 \u09b8\u09be\u09b0\u09bf\u09a4\u09c7 \u098f\u09ac\u0982 \u09b8\u09be\u09b0\u09bf\u0995\u09c7 \u0995\u09b2\u09be\u09ae\u09c7 \u09aa\u09b0\u09bf\u09a3\u09a4 \u0995\u09b0\u09c7 \u09b8\u09c3\u09b7\u09cd\u099f \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u0995\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995 <\/span>A<\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09c7 \u09af\u09c7 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df \u09a4\u09be\u0995\u09c7 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 <\/span>A<\/span> \u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09ac\u09b2\u09be \u09b9\u09df \u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 <\/span>A<\/span> \u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8<\/span> A^{-1}=\\frac{adj (A)}{|A|}\\frac{}{}<\/span><\/span><\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3 : A=\\begin{bmatrix}\n\n3 & -4 & 2\\\\\n\n-2 & 1 & 0\\\\\n\n-1 & -1 & 1\n\n\\end{bmatrix}<\/span><\/b>\u00a0\u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964\u00a0<\/b><\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8:<\/b><\/p>\n

A=\\begin{bmatrix}\n\n3 & -4 & 2\\\\\n\n-2 & 1 & 0\\\\\n\n-1 & -1 & 1\n\n\\end{bmatrix}<\/span> \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995 |A|=3(1+0)+4(-2+0)+2(2+1)=3-8+6=1<\/span><\/span><\/p>\nadj(A)=\\begin{bmatrix}\n\n1+0 & 0+2 & 2+1\\\\\n\n-2+4 & 3+2 & -4+3\\\\\n\n0-2 & 4-0 & 3-8\n\n\\end{bmatrix}^T-\\begin{bmatrix}\n\n1 & 2 & 3\\\\\n\n2 & 5 & -1\\\\\n\n-2 & -4 & -5\n\n\\end{bmatrix}^T=\\begin{bmatrix}\n\n1 & 2 & -2\\\\\n\n2 & 5 & -4\\\\\n\n3 & -1 & -5\n\n\\end{bmatrix}<\/span>\nA^{-1}=\\frac{1}{|A|}adj(A)=\\frac{1}{1}\\begin{bmatrix}\n\n1 & 2 & -2\\\\\n\n2 & 5 & -4\\\\\n\n3 & -1 & -5\n\n\\end{bmatrix}=\\begin{bmatrix}\n\n1 & 2 & -2\\\\\n\n2 & 5 & -4\\\\\n\n3 & -1 & -5\n\n\\end{bmatrix}<\/span>\n

 <\/p>\n

(iii) \u09ac\u09cd\u09b2\u0995 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (Block matrix method):<\/b><\/span><\/h3>\n

A=(a_{ij})_{n\\times n}<\/span>\u098f\u0995\u099f\u09bf \u0985\u09ac\u09cd\u09af\u09a4\u09bf\u0995\u09cd\u09b0\u09ae\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09b9\u09b2\u09c7 \u09ac\u09cd\u09b2\u0995 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 [A|I_n]<\/span><\/span> \u0995\u09c7 \u09b8\u09ae\u09a4\u09c1\u09b2\u09cd\u09af \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 [I_n|B]<\/span><\/span> \u09a4\u09c7 \u09aa\u09b0\u09bf\u09a3\u09a4 \u0995\u09b0\u09be \u09af\u09be\u09df, \u09af\u09c7\u0996\u09be\u09a8\u09c7 B=A^{-1}<\/span><\/span>\u00a0\u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3 :- A=\\begin{bmatrix}\n\n1 & 0 & 0\\\\\n\n0 & 1 & 0\\\\\n\n0 & 0 & 1\n\n\\end{bmatrix}<\/span><\/b>\u00a0\u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964\u00a0<\/b><\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8:\u00a0<\/strong>\u098f\u0996\u09be\u09a8\u09c7, [A|I]=\\left[\\begin{matrix}\n\n1 & 0 & 0 \\\\\n\n1& 0 & 1 \\\\\n\n1 & 1 & 0\n\n\\end{matrix}\\right|\\left.\\begin{matrix}\n\n1& 0 & 0\\\\\n\n0 & 1 & 0\\\\\n\n0 & 0 & 1\n\n\\end{matrix}\\right]\\xrightarrow[r_3-r_1]{r_2-r_1}\\left[\\begin{matrix}\n\n1 & 0 & 0 \\\\\n\n0& 0 & 1 \\\\\n\n0 & 1 & 0\n\n\\end{matrix}\\right|\\left.\\begin{matrix}\n\n1& 0 & 0\\\\\n\n-1 & 1 & 0\\\\\n\n-1 & 0 & 1\n\n\\end{matrix}\\right]\\xrightarrow[]{r_2-r_3} \\left[\\begin{matrix}\n\n1 & 0 & 0 \\\\\n\n0& 1 & 1 \\\\\n\n0 & 1 & 0\n\n\\end{matrix}\\right|\\left.\\begin{matrix}\n\n1& 0 & 0\\\\\n\n-1 & 0 & 1\\\\\n\n1 & -1 & 0\n\n\\end{matrix}\\right] \\xrightarrow[]{r_3-r_2} \\left[\\begin{matrix}\n\n1 & 0 & 0 \\\\\n\n0& 1 & 0 \\\\\n\n0 & 0 & -1\n\n\\end{matrix}\\right|\\left.\\begin{matrix}\n\n1& 0 & 0\\\\\n\n-2 & 1 & 1\\\\\n\n1 & -1 & 0\n\n\\end{matrix}\\right] \\xrightarrow[]{r_2-r_3} \\left[\\begin{matrix}\n\n1 & 0 & 0 \\\\\n\n0& 1 & 1 \\\\\n\n0 & 0 & -1\n\n\\end{matrix}\\right|\\left.\\begin{matrix}\n\n1& 0 & 0\\\\\n\n-1 & 0 & 1\\\\\n\n1 & -1 & 0\n\n\\end{matrix}\\right] \\xrightarrow[]{(-r_3)} \\left[\\begin{matrix}\n\n1 & 0 & 0 \\\\\n\n0& 1 & 0 \\\\\n\n0 & 0 & -1\n\n\\end{matrix}\\right|\\left.\\begin{matrix}\n\n1& 0 & 0\\\\\n\n-1 & 1 & 1\\\\\n\n1 & -1 & 0\n\n\\end{matrix}\\right]=[I|A^{-1}]<\/span><\/p>\n

\u2234A<\/span> \u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 A^{-1}=\\begin{bmatrix}\n\n1 & 0 & 0\\\\\n\n-1 & 0 & 1\\\\\n\n-1 & 1 & 0\n\n\\end{bmatrix}<\/span><\/span><\/p>\n

\u09b8\u09a4\u09cd\u09af\u09a4\u09be \u09af\u09be\u099a\u09be\u0987 : <\/b>\u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u0997\u09c1\u09a3\u09a8\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09df \u09af\u09c7, A.A^{-1}=A^{-1}.A=I_3<\/span><\/span><\/p>\n

\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u09c7\u09b0 \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af (Properties of Inverse Matrix)<\/b><\/span><\/h2>\n
    \n
  1. (A^{-1})^{-1}=A<\/span><\/span><\/li>\n
  2. (AB)^{-1}=B^{-1}A^{-1}<\/span><\/li>\n
  3. (A^T)^{-1}=(A^{-1})^T<\/span><\/li>\n
  4. (BA)A^{-1}=B(AA^{-1})=B<\/span><\/li>\n
  5. I=I^{-1}=I_n<\/span><\/li>\n
  6. AB=C\\Rightarrow A=CB^{-1} \u098f\u09ac\u0982 B=A^{-1}C<\/span><\/li>\n<\/ol>\n

    [MCQ \u098f\u09b0 \u099c\u09a8\u09cd\u09af: \\begin{bmatrix}\n\na & b\\\\\n\nc & d\n\n\\end{bmatrix}<\/span><\/strong><\/span>\u00a0\u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \\frac{1}{ad-bc}\\begin{bmatrix}\n\nd & -b\\\\\n\n-c & a\n\n\\end{bmatrix}\\begin{bmatrix}\n\na & 0 & 0\\\\\n\n0 & b & 0\\\\\n\n0 & 0 & c\n\n\\end{bmatrix}<\/span><\/span><\/p>\n

     <\/p>\n

    \n

     <\/p>\n

    \u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n