Let us find them using the elementary row operations on the coefficient matrix.ĭividing the 2 nd row by 51 and and 3 rd row by 17, Therefore, the system has an infinite number of solutions (along with the trivial solution (x, y, z) = (0, 0, 0)). Let us find the determinant of the coefficient matrix:
When t = 0.5: (x, y, z) = (-1, 0.5, 0.5), etcĮxample 3: How many solutions does the following system has? Find them all. For example, some nontrivial solutions of the above homogeneous system can be: Thus, the solution is (x, y, z) = (-2t, t, t) which represents an infinite number of nontrivial solutions as 't' can be replaced with one of the real numbers (which is an infinite set). Hence we should assume one of the variables to be a parameter (say t which is a real number). We have two equations in three variables. Just expand the first two rows of the above matrix as equations. It means that the system has nontrivial solutions also. We couldn't convert it into the upper diagonal matrix as we ended up with a row of zeros in the matrix. Let us take the coefficient matrix of the above system and apply row operations in order to convert it into an upper diagonal matrix.
We can find them using the matrix method and applying row operations. But it may (or may not) have other solutions than the trivial solutions that are called nontrivial solutions. For example, the system formed by three equations x + y + z = 0, y - z = 0, and x + 2y = 0 has the trivial solution (x, y, z) = (0, 0, 0). , 0) is obviously a solution to the system and is called the trivial solution (the most obvious solution). Since there is no constant term present in the homogeneous systems, (x₁, x₂. Solving Homogeneous System of Linear EquationsĪ homogeneous system may have two types of solutions: trivial solutions and nontrivial solutions.
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