ALGEBRA FOR STATISTICS — DU B.Sc (hons) Statistics Previous Year Question Papers

Attempt six questions in all, selecting three questions from each Section. (a) Define idempotent, involutory and trace of a matrix. Prove that: (b) If A is a non-singular matrix of order n, then show that : A mixture of weight 6 kilograms is to be made containing foods A, B and C. It is given that A, 8, C contains 500, ?00, 200 units of vitamins and 400, 600, 700 tinits of calories per kg. The mixture is to contain 1800 units of vitamins and 3500 units of calories. Using matric.es, find the composition of the mixture by forming `the necessary s.imultaneous linear equations. ` When is a rectangular matrix ,said to be in Ech61on form. ? Transform. the following matrix into an echelon form and encircle the pivot entries .: 3. (cz) Define circulant Determinant.. Prove that : Show that every square matrix cari be uniquely expressed as p + I.Q where P and Q are Hermitian matrices. (Zt) (cz) Comprite the inverse of the given matrix by the method of partitioning : where I is an identity matri2b P, Q, R are Su.b-matrices and it is given that the necessary ihverses exist. (a) .Express the determinait : 1 cos(a-or) cos(y-er) cos(ci -.P) 1 .cos(y -f}) cos(a.-y) cos(P-y) 1 as a product of two determinafits. Hence or` otherwise show that it vanishes. Define Vector spaces, Linear Dependence and Independence of vectors. Show that the vectors [1, 2, 4], [2, 2, 8], [1, 0, 4] are linearly dependent. Show that any two characteristic vectors corresponding to two distinct characteristic roots are 'orthogonal for the following matrices : (Z) Hermitian matrix, I (I.i) .Real symmetric matrix. (£a.Z) Uhitary matrix. Reduce the following symmetric matrix to the diagonal form and interpret the result in terms of quadratic forms : Find non-singular matrices R and s such that RAs is in normal form, where 7. (c!) .Hence find the rank of A. Define generalized inverse. Give an algorithm for finding a generalized i.nverse of a symmetric matrix- A. Hence find a g-inverse of (a) 8. (a) (a) The characteristic roots of a square matrix of order 3 are 1, T1, 2. Express A6 as the. quadratic pc;lynomial in A. Show that elementary matrices are nomsingular. Obtain their inveises. Show that the form : is indefinite and find sets of va.1ues of the variables #i, #2, #3 for which the given form assume§ positive, negative and zero- values.