Use MATLAB to construct a matrix A by setting A = vander(1 : 6); A = A − diag(sum(A...

Use MATLAB to construct a matrix A by setting

A = vander(1 : 6); A = A − diag(sum(A’))

(a) By construction, the entries in each row of A should all add up to zero. To check this, set x = ones(6, 1) and use MATLAB to compute the product Ax. The matrix A should be singular. Why? Explain. Use the MATLAB functions det and inv to compute the values of det(A) and A⁻¹. Which MATLAB function is a more reliable indicator of singularity?

(b) Use MATLAB to compute det(A^T ). Are the computed values of det(A) and det(A^T ) equal? Another way to check if a matrix is singular is to compute its reduced row echelon form. Use MATLAB to compute the reduced row echelon forms of A and A^T.

(c) To see what is going wrong, it helps to know how MATLAB computes determinants. The MATLAB routine for determinants first computes a form of the LU factorization of the matrix. The determinant of the matrix L is ±1, depending on whether an even or odd number of row interchanges were used in the computation. The computed value of the determinant of A is the product of the diagonal entries of U and det(L) = ±1. In the special case that the original matrix has integer entries, the exact determinant should take on an integer value. So in this case MATLAB will round its decimal answer to the nearest integer. To see what is happening with our original matrix, use the following commands to compute and display the factor U:

format short e

[ L , U ] = lu(A); U

In exact arithmetic, U should be singular. Is the computed matrix U singular? If not, what goes wrong? Use the following commands to see the rest of the computation of d = det(A):

format short

d = prod(diag(U))

d = round(d)