# Programming Assignment 3 – Recursive Calculation of the Determinant of a Matrix Write a program that uses a recursive algorithm to compute the determinant of a maxtrix. It should read a matrix, print it out, and compute and print the determinant. New: Redo assignment 2, but this time use linked lists to implement the array. Implement the matrix as a “sparse matrix” where entries with a value of zero do not take any space in memory. Determinant = Σ (-1)i+j * a[i,j]*det(minor(a[i,j]) for any j. i The minor of matrix element x is the submatrix formed by deleting the row and column containing x. For example, a= 1 3 -1 6 4 5 -2 -3 2 -1 0 4 9 7 -9 1 minor(a[2,3]) = 1 3 6 2 -1 4 9 7 1 For stopping cases, if “a” is a 1×1 matrix: a=[x], then det(a) = x. a b If “a” is a 2×2 matrix: a= c d, then det(a) = ad-bc The sample file input form is 4 4 0 0 -1 0 0 1 0 3 1 0 0 0 0 0 0 6 where the first line contains the dimensions (rows columns) of the matrix.

Recursive Calculation of the Determinant of a Matrix

Introduction:

The determinant of a matrix is an important concept in linear algebra. It is a scalar value that can be computed using various methods. In this programming assignment, we will be implementing a recursive algorithm to calculate the determinant of a matrix. The program should read a matrix, print it out, and compute and print the determinant.

Algorithm:

The recursive algorithm for calculating the determinant of a matrix involves breaking down the matrix into smaller submatrices. The determinant of a matrix is defined as the sum of the product of the elements of the matrix and the determinant of its corresponding minor matrices.

For a given matrix A, the general formula for calculating the determinant using recursion is as follows:

Determinant = Σ (-1)i+j * a[i,j] * det(minor(a[i,j]))

where i and j are the row and column indices of the matrix element, respectively.

The minor of a matrix element is the submatrix formed by deleting the row and column containing that element.

Stopping Cases:

The recursion process stops when the matrix A becomes a 1×1 matrix or a 2×2 matrix.

If A is a 1×1 matrix, i.e., a=[x], then the determinant det(A) is equal to x.

If A is a 2×2 matrix, i.e., a= [a b; c d], then the determinant det(A) can be calculated using the formula ad-bc.

Implementation:

To implement the recursive algorithm for computing the determinant of a matrix, we will need to read the matrix input from a file. The sample file input format is as follows:

4 4

0 0 -1 0

0 1 0 3

1 0 0 0

0 0 0 6

The first line of the file contains the dimensions (rows and columns) of the matrix.

We will represent the matrix using linked lists to implement the array. This will allow us to represent the matrix as a “sparse matrix” where entries with a value of zero do not take any space in memory. Linked lists provide an efficient way of storing and manipulating sparse matrices.

Conclusion:

In this programming assignment, we will be implementing a recursive algorithm to compute the determinant of a matrix. The algorithm involves breaking down the matrix into smaller submatrices and calculating the determinant of each submatrix recursively. We will represent the matrix using linked lists to implement the array and handle sparse matrices efficiently.

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