How To Find Non Trivial Solution Of A Matrix

Systems of Linear Equations

Introduction

Consider the two equations ax+by=c and dx+ey=f. Since these equations represent 2 lines in the xy-plane, the simultaneous solution of these two equations (i.e. those points (ten,y) that satisfy both equations) is but the intersection of the two lines. The graphs below illustrate the 3 possible cases: non-parallel lines, parallel (simply not identical) lines, and identical lines.

From left to right these cases yield i solution, no solutions, and infinite solutions. The same situation occurs in three dimensions; the solution of 3 equations with 3 unknowns is the intersection of the 3 planes.

There is a simple tool for determining the number of solutions of a square organisation of equations: the determinant.

Matrix Annotation

Information technology is often convenient to stand for a system of equations equally a matrix equation or even as a single matrix. Consider the arrangement of equations 2x+3y=-eight and -10+5y=1. Since

we may write the entire system equally a matrix equation:

or as AX=B where

The 2x2 matrix A is called the matrix of coefficients of the system of equations. Frequently this equation is written as a single augmented matrix:

We may now use Gaussian elimination to solve this matrix equation for 10 and y (as opposed to directly substitution of one equation into the other). This representation tin can too be done for any number of equations with whatever number of unknowns.

In general, the equation AX=B representing a organisation of equations is called homogeneous if B is the nx1 (cavalcade) vector of zeros. Otherwise, the equation is called nonhomogeneous.

Determining the Number of Solutions of a Nonhomogeneous System of Equations

For square systems of equations (i.e. those with an equal number of equations and unknowns), the most powerful tool for determining the number of solutions the organisation has is the determinant. Suppose we have 2 equations and two unknowns: ax+by=c and dx+ey=f with b and e non-zero (i.east. the system is nonhomogeneous). These are ii lines with slope -a/b and -d/eastward, respectively. Let's define the determinant of a 2x2 organisation of linear equations to be the determinant of the matrix of coefficients A of the system. For this system

Suppose this determinant is nil. Then, this last equation implies a/b=d/e; in other words, the slopes are equal. From our discussion above, this means the lines are either identical (there is an infinite number of solutions) or parallel (there are no solutions). If the determinant is non-goose egg, then the slopes must be dissimilar and the lines must intersect in exactly 1 point. This leads us to the post-obit result:

A nxn nonhomogeneous organization of linear equations has a unique non-trivial solution if and merely if its determinant is non-zip. If this determinant is zero, then the system has either no nontrivial solutions or an infinite number of solutions.

Example

The nonhomogeneous system of equations 2x+3y=-8 and -x+5y=ane has determinant

which implies the system has a unique solution (the signal (-43/xiii,-28/75)). This corresponds to our intuition: the lines have dissimilar slopes and therefore must intersect in exactly one point.

Example

The nonhomogeneous arrangement of equations x-4y+6z=3, -2x+8y-12z=-6, and 2x-y+3z=1 does not take a unique solution since its determinant is

In fact, this organization has an infinite number of solutions. To encounter this, think geometrically. The three equations represent 3 planes. Two of the equations (the start and the 2d) correspond the aforementioned aeroplane (why?); so, there are merely two singled-out non-parallel planes. Since 2 non-parallel planes intersect in a line, there are an space number of points which lie on all three of these planes (i.due east. the system has an infinite number of solutions). To actually find the solutions, we must use a method such equally Gaussian emptying to solve the arrangement of equations.

Determining the Number of Solutions of a Homogeneous Organization of Equations

For a homogeneous system of equations ax+past=0 and cx+dy=0, the state of affairs is slightly different. These lines pass through the origin. Thus, there is always at to the lowest degree one solution, the point (0,0). If the slopes -a/b and -c/d are equal and so at that place are an infinite number of solutions since the lines are identical. Merely as we have seen, the slopes of these lines are equal when the determinant of the coefficient matrix is nothing. Thus, for homogeneous systems we take the following result:

A nxn homogeneous system of linear equations has a unique solution (the piffling solution) if and simply if its determinant is non-zero. If this determinant is zero, then the organisation has an infinite number of solutions.