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 {
		   "cell_type": "markdown",
	   "metadata": {},
	   "source": [
       "# Chapter 5: Vectors and Matrices"
	   ]
	},
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.10: Determinants.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"A=[5 4;2 3];\n",
"det(A);  \n",
"disp(det(A),'determinant of A')\n",
"B=[2 1;-4 6]\n",
"det(B);\n",
"disp(det(B),'determinant of B')\n",
"C=[2 1 3;4 6 -1;5 1 0]\n",
"disp(det(C),'determinant of C')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.13: Matrix_solution_of_a_system_of_linear_equations.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"A=[1 2 1;2 5 -1;3 -2 -1];     //left hand side of the system of equations\n",
"B=[3 -4 5]';                   //right hand side or the constants in the equations\n",
"X=[];\n",
"X=A\B ;   //unique solution for the system of equations\n",
"x=X(1)\n",
"y=X(2)\n",
"z=X(3)"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.14: Inverse_of_a_square_matrix.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"A=[1 0 2;2 -1 3;4 1 8];\n",
"P=rref([A,eye(3,3)]);\n",
"disp(P,'canonical form of matrix A :')\n",
"disp('left side of the matrix P is the identity matrix so the right side is the inverse of A') \n",
"inverseA=P(:,4:6)  "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.2: Vector_operations.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"u=[2,3,-4];\n",
"v=[1,-5,8];\n",
"u+v\n",
"5*u\n",
"-v\n",
"2*u-3*v\n",
"u.*v;\n",
"k=sum(u.*v);\n",
"disp(k,'dot product of the two vectors')\n",
"l=norm(u); \n",
"disp(l,'norm or length of the vector u')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.3: Column_vectors.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"u=[5,3,-4]'\n",
"v=[3,-1,-2]'\n",
"2*u-3*v\n",
"k=sum(u.*v);\n",
"disp(k,'The dot product of the two vectors u and v is:')\n",
"l=norm(u);\n",
"disp(l,'The length or norm of the vector u is:')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.5: Matrix_addition_and_Scalar_multiplication.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"A=[1,-2,3;0,4,5];\n",
"B=[4,6,8;1,-3,-7];\n",
"k=A+B;\n",
"disp(k,'The addition of the two matrices A and B is:')\n",
"m=3*A;\n",
"disp(m,'The multiplication of a vector with a scalar is:')\n",
"p=2*A-3*B"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.6: Matrix_multiplicatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"a=[7,-4,5];\n",
"b=[3,2,-1]';\n",
"k=a*b;\n",
"disp(k,'product of a and b is;')\n",
"p=[6,-1,8,3];\n",
"q=[4,-9,-2,5]';\n",
"l=p*q;\n",
"disp(l,'product of p and q is:')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.7: Matrix_multiplicatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"A=[1 3;2 -1];\n",
"B=[2 0 -4;5 -2 6];\n",
"A*B\n",
"A=[1 2;3 4]\n",
"B=[5 6;0 -2];\n",
"A*B\n",
"B*A\n",
"disp('matrix mulitplication is not commutative since AB may not be equal to BA')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.8: Algebra_of_square_matrices.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"A=[1 2;3 -4];\n",
"A2=A*A  //multiplying A by itself\n",
"A3=A2*A\n",
"f=2*A2-3*A+5;  \n",
"disp(f,'for the function f(x)=2x^2-3x+5,f(A) is:')\n",
"g=A2+3*A-10;\n",
"disp(g,'for the function g(x)=x^2+3x-10,g(A) is')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 5.9: Invertible_matrices.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"A=[1 0 2;2 -1 3;4 1 8];\n",
"B=[-11 2 2;-4 0 1;6 -1 -1];\n",
"A*B\n",
"disp('since A*B is identity matrix,A and B are invertible and inverse of each other')"
   ]
   }
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