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document.write( "Q4: For the linear system \r\n" );
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document.write( "2x - y + 3z = 5\r\n" );
document.write( "3x + y + 4z = 2\r\n" );
document.write( "-x + 5y – 2z = -2 \r\n" );
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document.write( "(i)Write the system in matrix from AX = B \r\n" );
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document.write( " A X = B \r\n" );
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document.write( "[ 2 -1 3][x] [ 5]\r\n" );
document.write( "[ 3 1 4][y] = [ 2]\r\n" );
document.write( "[-1 5 –2][z] [-2] \r\n" );
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document.write( "(ii)Obtain the determinant and inverse of A. \r\n" );
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document.write( "I will assume you know how to get the value \r\n" );
document.write( "of a 3×3 determinant, and how to find the\r\n" );
document.write( "minors and the transpose. If you don't post \r\n" );
document.write( "again asking how.\r\n" );
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document.write( " | 2 -1 3| \r\n" );
document.write( "det(A) = | 3 1 4| = 2 \r\n" );
document.write( " |-1 5 –2| \r\n" );
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document.write( "First find the cofactor matrix.\r\n" );
document.write( "Replace each element of A by the value of\r\n" );
document.write( "its minor 2×2 determinant with the sign\r\n" );
document.write( "left as it is or changed according to the\r\n" );
document.write( "sign scheme:\r\n" );
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document.write( " |+ - +|\r\n" );
document.write( " |- + -|\r\n" );
document.write( " |+ - +|\r\n" );
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document.write( " [-22 2 16]\r\n" );
document.write( "cofactor matrix = [ 13 -1 -9]\r\n" );
document.write( " [ -7 1 5]\r\n" );
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document.write( "Take the transpose of the cofactor matrix,\r\n" );
document.write( "which is called the adjoint matrix:\r\n" );
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document.write( " [-22 13 -7]\r\n" );
document.write( "adjoint matrix = [ 2 -1 1]\r\n" );
document.write( " [ 16 -9 5]\r\n" );
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document.write( "Divide every member of the adjoint matrix\r\n" );
document.write( "by the determinant of A. This is the\r\n" );
document.write( "inverse of A, written A-1:\r\n" );
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document.write( " [-11 13/2 -7/2]\r\n" );
document.write( "Inverse matrix = A-1 = [ 1 -1/2 1/2]\r\n" );
document.write( " { 8 -9/2 5/2] \r\n" );
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document.write( "(iii)Solve the system.\r\n" );
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document.write( "Go back to the system in matrix form AX = B\r\n" );
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document.write( " A X = B\r\n" );
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document.write( " [ 2 -1 3][x] [ 5]\r\n" );
document.write( " [ 3 1 4][y] = [ 2]\r\n" );
document.write( " [-1 5 –2][z] [-2] \r\n" );
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document.write( "Left-multiply both sides by the inverse matrix and\r\n" );
document.write( "get the form A-1AX = A-1Bx\r\n" );
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document.write( " A-1 A X = A-1 X \r\n" );
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document.write( "[-11 13/2 -7/2] [ 2 -1 3][x] [-11 13/2 -7/2][ 5]\r\n" );
document.write( "[ 1 -1/2 1/2] [ 3 1 4][y] = [ 1 -1/2 1/2][ 2]\r\n" );
document.write( "{ 8 -9/2 5/2] [-1 5 –2][z] { 8 -9/2 5/2][-2]\r\n" );
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document.write( "Do the matrix multiplication A-1AX = A-1B becomes\r\n" );
document.write( " IX = A-1B which becomes:\r\n" );
document.write( " X = A-1B\r\n" );
document.write( " like this:\r\n" );
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document.write( " I X = A-1B \r\n" );
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document.write( " [1 0 0][x] [-35]\r\n" );
document.write( " [0 1 0][y] = [ 3]\r\n" );
document.write( " [0 0 1][z] [ 26]\r\n" );
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document.write( " X = A-1B \r\n" );
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document.write( " [x] [-35]\r\n" );
document.write( " [y] = [ 3]\r\n" );
document.write( " [z] [ 26]\r\n" );
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document.write( "Edwin \n" );
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