Mathematics Assignment - MATRIX ALGEBRA AND DETERMINANTS

Q.1: If A and B are two square matrices of order ‘n', then show that (Recurrence Relations)

i) (A + B)² = A² + B² + AB + BA                    ii) (A - B)² = A² + B² - AB - BA
iii) (A - B)(A + B) = A² - B² - BA + AB             iv) (A + B)(A - B) = A² - B² + BA - AB

Q.2: If E = and F = , compute the matrix products

i) EF    ii) FE     iii) E²F +F²E     iv) and prove that (E + xF²)³ = x I, where x being a scalar and I being identity matrix.

Q.3: For matrices show that

v) The product AB of two Hermitian matrices A and B, is Hermitian matrix iff A and B commute.

vi) The product AB of two symmetric matrices A and B, is a symmetric matrix iff A and Bcommute.

vii) If A and B are square matrices of order 2 such that then show that (A - B) and (A + B) anti-commute.

viii) If A & B are symmetric (anti-symmetric), then A+B is symmetric (anti-symmetric).

ix) Every square matrix is uniquely expressed as a P+iQ where P and Q are Hermitian matrices.

Q.4: Prove that

iii) The three 2x2 matrices E₁, E₂, E₃, satisfying the relations

E_iE_j + E_jE_i = 0      for i≠j, 1 ≤ i, j ≤ 3

E_i² = -I              for 1 ≤ i ≤ 3

Where I is the identity matrix. If A = x₀I + x₁E₁ + x₂E₂ + x₃E₃ where x₀, x₁, x₂, and x₃ are the non-zero scalars, then prove that

A(2x₀ I - A) = ( x₀² + x₁² + x₂² + x₃²)I

iv) If e^A is defined as I + A + A²/2! +......, show that

v) If A = , prove that (I + A) = (I - A)

vi) If ω is the cube root of unity, prove that a + bω + cω² is a factor of . Hence evaluate the determinant.

Q.5:i) If A = find out the values of α and β such that (αI + βA)² = A. Where I is the identity matrix.

ii) If P = [a_ij] and Q = [b_ij] where I, j = 1, 2, 3 and b_ij = 2^i+ja_ij. If |p| = 2 then find out det(Q).

Q.6: Find the adjoint of the matrix A = and verify that

A. (adj A) = (adj A).A = |A|.I

Q.7: Find the reciprocal of A = by the method of elementary operations.

Q.8: Find the reciprocal of the matrix S = and show that the transform of the matrix

by S, i.e., SAS^-1 is a diagonal matrix.

Q.9: Prove that 

Q.10: Find the rank and nullity of matrix A, where A = .