iasc said:The question is:
Compute the determinant of the n × n matrix A for which aij = max(i, j).
I can compute determinants but I don't really know what the last bit means.
Any help appreciated.
The determinant of a matrix is a mathematical concept that helps us understand the properties and behavior of linear transformations. It also plays a crucial role in solving systems of linear equations and finding the inverse of a matrix.
To compute the determinant of a matrix, you can use various methods such as cofactor expansion, Gaussian elimination, or using the properties of determinants. The method you choose may depend on the size and complexity of the matrix.
In this context, "max(i,j)" means that we are only considering the terms in the matrix where the row number (i) is less than or equal to the column number (j). This notation is used to simplify the computation of the determinant in certain cases.
Yes, the determinant of a matrix can be negative. The sign of the determinant depends on the number of row switches needed to transform the matrix into its reduced row echelon form. An odd number of row switches will result in a negative determinant, while an even number of row switches will result in a positive determinant.
The determinant of a matrix has various applications in mathematics, physics, engineering, and computer science. It is used to solve systems of linear equations, find the inverse of a matrix, calculate volumes of parallelepipeds and determine the stability of a dynamic system, among other things.