What happens if determinant is zero

Skip to main content. Search for:. Understanding Properties of Determinants There are many properties of determinants.

A General Note: Properties of Determinants If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal. When two rows are interchanged, the determinant changes sign. If either two rows or two columns are identical, the determinant equals zero.

That means x and y vectors do not form an area. Hence, the det A is zero. Det refers to the area formed by the vectors. In plain English then, if a matrix is invertible then it may have a solution. If a matrix's determinant is nonzero, the matrix may have a solution. If the determinant is zero, then the matrix is not invertible and thus does not have a solution because one of the rows can be eliminated by matrix substitution of another row in the matrix.

Common reasons for matrix invertibility are that one or more rows in the matrix is a scalar of the other. You see that Row 3 can be duplicated by adding Row 1 and Row 2. In short, if the determinant of a matrix is zero, the matrix does not have a solution because the matrix cannot be inverted.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. What does it mean to have a determinant equal to zero? Ask Question. Asked 8 years, 7 months ago. Active 1 year, 11 months ago. Viewed k times. I hope someone can explain this to me in plain English. Rodrigo de Azevedo If the rank of an nxn matrix is smaller than n, the determinant will be zero.

Add a comment. Active Oldest Votes. The system of homogenous linear equations represented by the matrix has a non-trivial solution. Ittay Weiss Ittay Weiss I'd never thought about that.

Show 2 more comments. When the determinant of a matrix is zero, the volume of the region with sides given by its columns or rows is zero, which means the matrix considered as a transformation takes the basis vectors into vectors that are linearly dependent and define 0 volume. The determinant of a matrix is a special value that is calculated from a square matrix. It can help you determine whether a matrix has an inverse, find the area of a triangle, and let you know if the system of equations has a unique solution.

Determinants are also used in calculus and linear algebra. Determinants are defined only for square matrices. If the determinant of a matrix is 0, the matrix is said to be singular, and if the determinant is 1, the matrix is said to be unimodular. If the determinant is nonzero than there exists exactly one solution.

If the determinant is zero, there could be no solutions, or there could be infinitely many. For no solution, Two lines have no solution, if these two lines are parallel to each other.

The lines are parallel to each other means that the slopes of the lines are equal. Sometimes that means that every single number is a solution, and sometimes it just means all the numbers that fit a certain pattern. No solution would mean that there is no answer to the equation.