By N. Hitchin
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Extra resources for Molopoles, Minimal Surfaces and Algebraic Curves
4. 22) has no solution. 5 Linear Systems of Equations . 5. 23) has infinitely many solutions. A system of simultaneous linear equations is consistent if it possesses at least one solution. If no solution exists, the system is inconsistent. 22) in inconsistent. The graph of a linear equation in three variables is a plane in space; hence a system of linear equations in three variables is depicted graphically by a set of planes. A solution to such a system is the set of coordinates for a point in space that lies on all the planes defined by the equations.
24) is a test for checking whether one matrix is an inverse of another matrix. 6, we prove that if AB ¼ I for two square matrices of the same order, then A and B commute under multiplication and BA ¼ I. If we borrow this result, we reduce the checking procedure by half. A square matrix B is an inverse of a square matrix A if either AB ¼ I or BA ¼ I; each equality guarantees the other. We also show later in this section that an inverse is unique; that is, if a square matrix has an inverse, it has only one.
0 0 0 . . lk 3 7 7 7 7 is 7 5 2 À1 D 1=l1 6 0 6 6 ¼6 0 6 .. 4 . 0 0 1=l2 0 .. 0 0 1=l3 .. 0 0 ... ... . 0 0 0 .. . . 1=lk 3 7 7 7 7 7 5 50 . Matrices if none of the diagonal elements is zero. It is easy to show that if any diagonal element in a diagonal matrix is zero, then that matrix is singular (see Problem 56). An elementary matrix E is a square matrix that generates an elementary row operation on a matrix A under the multiplication EA. An elementary matrix E is a square matrix that generates an elementary row operation on a matrix A (which need not be square) under the multiplication EA.
Molopoles, Minimal Surfaces and Algebraic Curves by N. Hitchin