Determinant of matrix with Aij = min(i, j)

Therefore, the determinant of the new matrix is equal to the determinant of the original matrix. In summary, the determinant of a n x n matrix where each entry is the smaller value between its row and column index is always 1, and this can be proven by using elementary row operations to transform the matrix into a lower triangular matrix.
  • #1
nedf
4
0
Given a n x n matrix whose (i,j)-th entry is i or j, whichever smaller, eg.
[1, 1, 1, 1]
[1, 2, 2, 2]
[1, 2, 3, 3]
[1, 2, 3, 4]
The determinant of any such matrix is 1.
How do I prove this? Tried induction but the assumption would only help me to compute the term for Ann mirror.
 
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  • #2
From each line subtract the line immediately above it.
 
  • #3
I will get this?
[1 1 1 1]
[0 1 1 1]
[0 0 1 1]
[0 0 0 1]
How do I prove that this matrix has same determinant as the original one?
 
  • #4
nedf said:
I will get this?
[1 1 1 1]
[0 1 1 1]
[0 0 1 1]
[0 0 0 1]
Yes.

nedf said:
How do I prove that this matrix has same determinant as the original one?
Subtracting another line is one of the elementary row operations and is known to preserve the determinant.
 

FAQ: Determinant of matrix with Aij = min(i, j)

What is the "Determinant of matrix with Aij = min(i, j)"?

The "Determinant of matrix with Aij = min(i, j)" is a mathematical operation used to calculate the value of a square matrix with elements defined by the minimum value between the row and column indices. It is denoted as |A| or det(A).

How is the "Determinant of matrix with Aij = min(i, j)" calculated?

The determinant can be calculated by using the Laplace expansion method or by using the properties of determinants, such as row operations and cofactor expansion. The formula for a 3x3 matrix is: |A| = a11(a22a33 − a23a32) - a12(a21a33 − a23a31) + a13(a21a32 − a22a31).

What is the significance of the "Determinant of matrix with Aij = min(i, j)"?

The determinant is a useful mathematical tool that has many applications, including solving systems of linear equations, calculating the inverse of a matrix, and determining the area or volume of a parallelogram or parallelepiped. It also gives information about the linear transformation represented by the matrix, such as whether it preserves or reverses orientation.

Can the "Determinant of matrix with Aij = min(i, j)" be negative?

Yes, the determinant can be positive, negative, or zero. The sign of the determinant depends on the arrangement of the elements in the matrix and the properties of the matrix. For example, if the matrix has an odd number of negative elements, the determinant will be negative.

Is there a shortcut to calculate the "Determinant of matrix with Aij = min(i, j)"?

Unfortunately, there is no shortcut or direct formula to calculate the determinant of a matrix with Aij = min(i, j). The process of calculating the determinant involves a series of calculations and manipulations of the elements in the matrix. However, there are certain properties and rules that can make the calculation easier and more efficient.

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