1. Let 0 1 A 0 0 = , show that (aI + bA)n = an I + nan 1 bA, where I is the identity matrix of order 2 and n N. 2. If 1 1 1 A 1 1 1 1 1 1 = , prove that An n n n n n n n n n = n 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 , N. 3. If 3 4 1 2 4 A , then prove that A 1 1 1 2 n n n n n + = = , where n is any positive integer. 4. If A and B are symmetric matrices, prove that AB BA is a skew symmetric matrix. 5. Show that the matrix BAB is symmetric or skew symmetric according as A is symmetric or skew symmetric. 6. Find the values of x, y, z if the matrix 0 2 A y z x y z x y z = satisfy the equation AA = I. 7. For what values of x : [ ] 1 2 0 0 1 2 1 2 0 1 2 1 0 2 x = O? 8. If 3 1 A 1 2 = , show that A2 5A + 7I = 0. 9. F
Question 1:Let 0 1 A 0 0 = , show that (aI + bA)n = an I + nan – 1 bA, where I is the identity matrix of order 2 and n Î N.
Question 2:If 1 1 1 A 1 1 1 1 1 1 = , prove that An n n n n n n n n n = n é ë êêê ù û úúú Î − − − − − − − − − 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 , N.
Question 3:If 3 4 1 2 4 A , then prove that A 1 1 1 2 n n n n n − + − = = − − , where n is any positive integer.
Question 4:If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
Question 5:Show that the matrix B¢AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Question 6:Find the values of x, y, z if the matrix 0 2 A y z x y z x y z = − − satisfy the equation A¢A = I.
Question 7:For what values of x : [ ] 1 2 0 0 1 2 1 2 0 1 2 1 0 2 x = O?
Question 8:If 3 1 A 1 2 = − , show that A2 – 5A + 7I = 0.
Question 9:Find x, if [ ] 1 0 2 5 1 0 2 1 4 O 2 0 3 1 x x − − =
Question 10:A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below: Market Products I 10,000 2,000 18,000 II 6,000 20,000 8,000 (a) If unit sale prices of x, y and z are ` 2.50, ` 1.50 and ` 1.00, respectively, find the total revenue in each market with the help of matrix algebra. (b) If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.
Question 11:Find the matrix X so that 1 2 3 7 8 9 X 4 5 6 2 4 6 − − − =
Question 12:If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABn = BnA. Further, prove that (AB)n = AnBn for all n Î N. Choose the correct answer in the following questions:
Question 13:If A = a b g −a é ë ê ù û ú is such that A² = I, then (A) 1 + a² + bg = 0 (B) 1 – a² + bg = 0 (C) 1 – a² – bg = 0 (D) 1 + a² – bg = 0
Question 14:If the matrix A is both symmetric and skew symmetric, then (A) A is a diagonal matrix (B) A is a zero matrix (C) A is a square matrix (D) None of these
Question 15:If A is square matrix such that A2 = A, then (I + A)³ – 7 A is equal to (A) A (B) I – A (C) I (D) 3A
