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.
If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero. Non-vertical parallel lines have the same slopes! The slopes are equal. Since slope is a measure of the angle of a line from the horizontal, and since parallel lines must have the same angle, then parallel lines have the same slope — and lines with the same slope are parallel. Users questions. What does a determinant of 0 mean? What is the point of a determinant? What does it mean if determinant is 1?
How many solutions if determinant is zero? What is the condition of no solution? What are infinitely many solutions? What is the difference between no solution and infinitely many solutions?
How do you know if a system has no solution? This is more expressed in 3-Dimensions. The sign of the value of determinant gives a kind of information on the orientation of this body. If the determinant is zero, this means the volume is zero. This can only happen when one of the vectors "overlaps" one of the others or more formally, when two of the vectors or linearly dependent.
When you think a matrix as transformation, the determinant is the area or volume in higher dimension made by basis after transformation. This type of thinking will give you visual aid. The idea can be generalized for higher dimension. Note that, it is not possible to map back a line to a plane.
So, inverse of a matrix, which has determinant zero, doesn't exists. If you plot that, you can see that they are in the same span.
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.
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