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"source": [
"# Chapter 5 - Determinants"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 143 Example 5.3"
]
},
{
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"execution_count": 14,
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{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \n",
"[[ 3. 9.]\n",
" [ 3. 1.]]\n",
"D1(A) = 3.0\n",
"D2(A) = -27.0\n",
"D(A) = D1(A) + D2(A) = -24.0\n",
"That is, D is a 2-linear function.\n"
]
}
],
"source": [
"import numpy as np\n",
"A = np.random.rand(2,2)\n",
"for x in range(0,2):\n",
" for y in range(0,2):\n",
" A[x,y]=round(A[x,y]*10)\n",
"print 'A = \\n',A\n",
"\n",
"D1 = A[0,0]*A[1,1]\n",
"D2 = - A[0,1]*A[1,0]\n",
"print 'D1(A) = ',D1\n",
"print 'D2(A) = ',D2\n",
"print 'D(A) = D1(A) + D2(A) = ',D1 + D2\n",
"print 'That is, D is a 2-linear function.'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 145 Example 5.4"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \n",
"Matrix([[x, 0, -x**2], [0, 1, 0], [1, 0, x**3]])\n",
"e1,e2,e3 are the rows of 3*3 identity matrix, then\n",
"e1 = [ 1. 0. 0.]\n",
"e2 = [ 0. 1. 0.]\n",
"e3 = [ 0. 0. 1.]\n",
"D(A) = D(x*e1 - x**2*e3, e2, e1 + x**3*e3)\n",
"Since, D is linear as a function of each row,\n",
"D(A) = x*D(e1,e2,e1 + x**3*e3) - x**2*D(e3,e2,e1 + x**3*e3)\n",
"D(A) = x*D(e1,e2,e1) + x**4*D(e1,e2,e3) - x**2*D(e3,e2,e1) - x**5*D(e3,e2,e3)\n",
"As D is alternating, So\n",
"D(A) = (x**4 + x**2)*D(e1,e2,e3)\n"
]
}
],
"source": [
"import numpy as np\n",
"import sympy as sp\n",
"\n",
"x = sp.Symbol(\"x\")\n",
"A = sp.Matrix(([x, 0, -x**2],[0, 1, 0],[1, 0, x**3]))\n",
"print 'A = \\n',A\n",
"print 'e1,e2,e3 are the rows of 3*3 identity matrix, then'\n",
"T = np.identity(3)\n",
"e1 = T[0,:]\n",
"e2 = T[1,:]\n",
"e3 = T[2,:]\n",
"print 'e1 = ',e1\n",
"print 'e2 = ',e2\n",
"print 'e3 = ',e3\n",
"print 'D(A) = D(x*e1 - x**2*e3, e2, e1 + x**3*e3)'\n",
"print 'Since, D is linear as a function of each row,'\n",
"print 'D(A) = x*D(e1,e2,e1 + x**3*e3) - x**2*D(e3,e2,e1 + x**3*e3)'\n",
"print 'D(A) = x*D(e1,e2,e1) + x**4*D(e1,e2,e3) - x**2*D(e3,e2,e1) - x**5*D(e3,e2,e3)'\n",
"print 'As D is alternating, So'\n",
"print 'D(A) = (x**4 + x**2)*D(e1,e2,e3)'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 147 Example 5.5"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \n",
"Matrix([[x - 1, x**2, x**3], [0, x - 2, 1], [0, 0, x - 3]])\n",
"\n",
"E(A) = x**3 - 6*x**2 + 11*x - 6\n",
"--------------------------------------\n",
"A = \n",
"Matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])\n",
"\n",
"E(A) = 1\n"
]
}
],
"source": [
"import sympy as sp\n",
"\n",
"#part a\n",
"x = sp.Symbol('x')\n",
"A = sp.Matrix(([x-1, x**2, x**3],[0, x-2, 1],[0, 0, x-3]))\n",
"print 'A = \\n',A\n",
"print '\\nE(A) = ',A.det()\n",
"print '--------------------------------------'\n",
"#part b\n",
"A = sp.Matrix(([0 ,1, 0],[0, 0, 1],[1 ,0, 0]))\n",
"print 'A = \\n',A\n",
"print '\\nE(A) = ',A.det()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 158 Example 5.6"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Given Matrix:\n",
"A = \n",
"[[ 1 -1 2 3]\n",
" [ 2 2 0 2]\n",
" [ 4 1 -1 -1]\n",
" [ 1 2 3 0]]\n",
"After, Subtracting muliples of row 1 from rows 2 3 4\n",
"R2 = R2 - 2*R1\n",
"R3 = R3 - 4*R1\n",
"R4 = R4 - R1\n",
"A = \n",
"[[ 1 -1 2 3]\n",
" [ 0 4 -4 -4]\n",
" [ 0 5 -9 -13]\n",
" [ 0 3 1 -3]]\n",
"We obtain the same determinant as before.\n",
"Now, applying some more row transformations as:\n",
"R3 = R3 - 5/4 * R2\n",
"R4 = R4 - 3/4 * R2\n",
"We get B as:\n",
"B = \n",
"[[ 1 -1 2 3]\n",
" [ 0 4 -4 -4]\n",
" [ 0 0 -4 -8]\n",
" [ 0 0 4 0]]\n",
"Now,determinant of A and B will be same\n",
"det A = det B = 128.0\n"
]
}
],
"source": [
"import numpy as np\n",
"\n",
"print 'Given Matrix:'\n",
"A = np.array([[1, -1, 2, 3],[ 2, 2, 0, 2],[ 4, 1 ,-1, -1],[1, 2, 3, 0]])\n",
"print 'A = \\n',A\n",
"print 'After, Subtracting muliples of row 1 from rows 2 3 4'\n",
"print 'R2 = R2 - 2*R1'\n",
"A[1,:] = A[1,:] - 2 * A[0,:]\n",
"print 'R3 = R3 - 4*R1'\n",
"A[2,:] = A[2,:] - 4 * A[0,:]\n",
"print 'R4 = R4 - R1'\n",
"A[3,:] = A[3,:] - A[0,:]\n",
"print 'A = \\n',A\n",
"T = A# #Temporary matrix to store A\n",
"print 'We obtain the same determinant as before.'\n",
"print 'Now, applying some more row transformations as:'\n",
"print 'R3 = R3 - 5/4 * R2'\n",
"T[2,:] = T[2,:] - 5./4 * T[1,:]\n",
"print 'R4 = R4 - 3/4 * R2'\n",
"T[3,:] = T[3,:] - 3./4 * T[1,:]\n",
"B = T#\n",
"print 'We get B as:'\n",
"print 'B = \\n',B\n",
"print 'Now,determinant of A and B will be same'\n",
"print 'det A = det B = ',np.linalg.det(B)\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 160 Example 5.7"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \n",
"Matrix([[x**2 + x, x + 1], [x - 1, 1]])\n",
"B = \n",
"Matrix([[x**2 - 1, x + 2], [x**2 - 2*x + 3, x]])\n",
"det A = x + 1\n",
"det B = -6\n",
"Thus, A is not invertible over K whereas B is invertible\n",
"adj A = Matrix([[(x + 1)*((x - 1)/(x*(x + 1)) + 1/(x*(x + 1))), -x - 1], [-x + 1, x*(x + 1)]])\n",
"adj B = Matrix([[-6*(x + 2)*(-x**2/6 + 1/6)*(x**2 - 2*x + 3)/(x**2 - 1)**2 - 6/(x**2 - 1), 6*(x + 2)*(-x**2/6 + 1/6)/(x**2 - 1)], [6*(-x**2/6 + 1/6)*(x**2 - 2*x + 3)/(x**2 - 1), x**2 - 1]])\n",
"(adj A)A = (x+1)I\n",
"(adj B)B = -6I\n",
"B inverse = Matrix([[(x + 2)*(-x**2/6 + 1/6)*(x**2 - 2*x + 3)/(x**2 - 1)**2 + 1/(x**2 - 1), -(x + 2)*(-x**2/6 + 1/6)/(x**2 - 1)], [-(-x**2/6 + 1/6)*(x**2 - 2*x + 3)/(x**2 - 1), -x**2/6 + 1/6]])\n"
]
}
],
"source": [
"import sympy as sp\n",
"import numpy as np\n",
"\n",
"x = sp.Symbol(\"x\")\n",
"A = sp.Matrix(([x**2+x, x+1],[x-1, 1]))\n",
"B = sp.Matrix(([x**2-1, x+2],[x**2-2*x+3, x]))\n",
"print 'A = \\n',A\n",
"print 'B = \\n',B\n",
"print 'det A = ',A.det()\n",
"print 'det B = ',B.det()\n",
"print 'Thus, A is not invertible over K whereas B is invertible'\n",
"\n",
"print 'adj A = ',(A**-1)*A.det()\n",
"print 'adj B = ',(B**-1)*B.det()\n",
"print '(adj A)A = (x+1)I'\n",
"print '(adj B)B = -6I'\n",
"print 'B inverse = ',B**-1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 161 Example 5.8"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \n",
"[[1 2]\n",
" [3 4]]\n",
"det A = Determinant of A is: -2.0\n",
"Adjoint of A is: [[-2. -0. ]\n",
" [-0. -0.5]]\n",
"Thus, A is not invertible as a matrix over the ring of integers.\n",
"But, A can be regarded as a matrix over field of rational numbers.\n",
"Then, A is invertible and Inverse of A is: [[-2. 1. ]\n",
" [ 1.5 -0.5]]\n"
]
}
],
"source": [
"import numpy as np\n",
"\n",
"A = np.array([[1, 2],[3, 4]])\n",
"print 'A = \\n',A\n",
"d = np.linalg.det(A)#\n",
"print 'det A = ','Determinant of A is:',d\n",
"\n",
"\n",
"ad = (d* np.identity(2)) / A\n",
"print 'Adjoint of A is:',ad\n",
"\n",
"\n",
"print 'Thus, A is not invertible as a matrix over the ring of integers.'\n",
"print 'But, A can be regarded as a matrix over field of rational numbers.'\n",
"In = np.linalg.inv(A)#\n",
"#The A inverse matrix given in book has a wrong entry of 1/2. It should be -1/2.\n",
"print 'Then, A is invertible and Inverse of A is:',In\n"
]
}
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