True or FalseIf x is a nonzero vector in Rn and Ax = 0, thendet(A) = 0

Generate random 55 matrices with integer entries by setting A = round(10 rand(5)) and B = round(20 rand(5)) 10Use MATLAB to compute each of the pairs of numbers that follow. In each case, check whether the first number is equal to the second. (a) det(A) det(AT ) (b) det(A + B) det(A) + det(B) (c) det(AB) det(A) det(B) (d) det(ATBT ) det(AT ) det(BT ) (e) det(A1) 1/ det(A) (f) det(AB1) det(A)/ det(B)

Are n n magic squares nonsingular? Use the MATLAB command det(magic(n)) to compute the determinants of the magic squares matrices in the cases n = 3, 4, ... , 10. What seems to be happening? Check the cases n = 24 and 25 to see if the pattern still holds.

Set A = round(10 rand(6)). In each of the following, use MATLAB to compute a second matrix as indicated. State how the second matrix is related to A and compute the determinants of both matrices. How are the determinants related? (a) B = A; B(2, :) = A(1, :); B(1, :) = A(2, :) (b) C = A; C(3, :) = 4 A(3, :) (c) D = A; D(5, :) = A(5, :) + 2 A(4, :)

We can generate a random 6 6 matrix A whose entries consist entirely of 0s and 1s by setting A = round(rand(6)) (a) What percentage of these random 01 matrices are singular? You can estimate the percentage in MATLAB by setting y = zeros(1, 100); and then generating 100 test matrices and setting y(j) = 1 if the jth matrix is singular and 0 otherwise. The easy way to do this in MATLAB is to use a for loop. Generate the loop as follows: for j = 1 : 100 A = round(rand(6)); y(j) = (det(A) == 0); end (Note: A semicolon at the end of a line suppresses printout. It is recommended that you include one at the end of each line of calculation that occurs inside a for loop.) To determine how many singular matrices were generated, use the MATLAB command sum(y). What percentage of the matrices generated were singular? (b) For any positive integer n, we can generate a random 6 6 matrix A whose entries are integers from 0 to n by setting A = round(n rand(6)) What percentage of random integer matrices generated in this manner will be singular if n = 3? If n = 6? If n = 10? We can estimate the answers to these questions by using MATLAB. In each case, generate 100 test matrices and determine how many of the matrices are singular

If a matrix is sensitive to roundoff errors, the computed value of its determinant may differ drastically from the exact value. For an example of this, set U = round(100 rand(10)); U = triu(U, 1) + 0.1 eye(10) In theory, det(U) = det(UT ) = 1010 and det(UUT ) = det(U) det(UT ) = 1020 Compute det(U), det(U ), and det(U U ) with MATLAB. Do the computed values match the theoretical values?

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 A1. Which MATLAB function is a more reliable indicator of singularity? (b) Use MATLAB to compute det(AT ). Are the computed values of det(A) and det(AT ) 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 AT . (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. 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))

True or FalseA triangular matrix is nonsingular if and only if its diagonal entries are all nonzero.

True or FalseIf x is a nonzero vector in Rn and Ax = 0, then det(A) = 0.

True or FalseIf A and B are row equivalent matrices, then their determinants are equal

True or FalseIf A = O, but Ak = O (where O denotes the zero matrix) for some positive integer k, then A must be singular.