1. Write the following matrix as a sum of a symmetric and a skew-symmetric matrix 4 5 6 A= 65 5 6 4 [5] 2. Show the product of the matrices B and C, where 2 3) 4 0 0 B= 3 2 0 and C= 5 6 0 [5] 20 0 6 5 4 3. Find the determinant and Hence the inverse of the matrix D, where 10 3 D = 0 2 4 [14] 0 03 4. Using the concept of echelon form (row operations), find the rank of the following matrix E, where 2 3 4 [11] 1 1 5 E = 1 3 4 2 3 5 9 5 . Using the Cramer's rule to find a unique solution of the system of linear equations xty+ z=4, x+ 2y +3z = 8 and x + 2y +4z =9. [15]1. Write the pattern of the matrix (symmetric, skew-symmetric or none of them) with details 0 2 3 [5] A=-2 0 4 -3 -4 0 2. Find the matrix B, when 2A +3B =41, , where the matrix A is 0 2 3 A = -2 0 4 [6] -3 -4 0 3. Show the product of the matrices Band C, where 1 2 1 (1 0 0) B = 1 2 0 and C= 2 1 0 [7] 20 0 2 2 : 4. Find the adjoint, determinant and Hence the inverse of the matrix D, where 1 1 3 D= 0 2 : [16] 0 35. Using the Gauss Elimination method, find the value of 1 and a so that the equations x + 2y + 3z = 2, -x + 3y + 7z = a and 2x + 9y + Xz = a have (i) unique solution [16] (ii) no solution and infinitely many solution.1. Find the Eigen values and Eigen vectors of the matrix A, where 4- ( 2 3) [12] 2. Find all third roots of the complex number z, where z =1-i. [6] 3. Find the inverse of the Laplace transform of F(s), where F (s) = In(7+s [7] 4. Verify, if the Laplace transform of the function 1 - cost f (t) = t Exists or not. If exist, then find the Laplace transform of the function f (t). [10] Note that : x2 lim = 1 1 1+x2 5. Solve the initial value problem using the Laplace transform method y"+ y=1, y(0) = 0, y'(0) =1. [15]