MAT212 Worksheet PDF

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for first part of the course which is elementary to matrices...

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MAT 212 (2019) Worksheet 1 QUESTION 1 (a) Show that A + AT is symmetric for any square matrix A. (b) Show that if A is skew symmetric then A2 is symmetric. (c) A matrix P is idempotent if P 2 = P . Show that if P is idempotent then P T is idempotent. What are the possible values of det(P )?Ans. det(P ) = 0, 1 QUESTION 2 (a) Find a, b, c and d if     1 1 a−b b−c =2 . c−d d−a −3 1 S = {(a, b, c, d) = (4, 2, 0, 6)} (b) Show that if A is a square matrix such that A = A−1 then det(A) = ±1. (c) Show that if B is invertible, then det(B −1 AB) = det(A). (d) Find A when (In − 2AT )−1 =

  2 1 . 1 1

  1 0 1 A= . 2 1 −1 QUESTION 3 (a) Use Companion method for the inverse matrix to solve the following system of linear equations by finding the inverse of the coefficient matrix. −x − y = 1 2x + 5y + 5z = 0 2x + 4y + 3z = −1 Sol. x = 2, y = −1, z = 1

(b) Find all the values of k for which   k −k 3 1  A = 0 k + 1 k −8 k − 1 is invertible. Ans:k = 6 0, k 6= 2     a b c  a + x b + y c + z     3y 3z  = 18 (c) Show that if p q r  = 6, then  3x x y z   −p −q −r    1 1 1   (d) Show that  a b c  = (a + b + c)(a − b)(c − a) a3 b3 c3

QUESTION 4 Consider the following matrices 

0 −1 3 1 0 −2 6 1 A= 0 3 −9 2 0 1 −3 −1  1 −2 0 2 1 0 0 1 5 0 B=  0 0 0 0 1 0 0 0 0 0

3 −5 4 3 1 −3 6 0

 2 1 0 −1   1 −1 0 1  1 −1 . 1 0

(a) Reduce A to its reduced row echelon form. (b) Find the general solution of the linear system of which the augmented matrix has been reduced to B . (a)  0 0 B = 0 0

1 −3 0 0 0 0 0 0

0 1 0 0

0 0 1 0

 0 0 0 −1  . 0 0  1 1

(b) S = {(2r − 2s − t + 1, r, −5s + 3t − 1, s, −6t + 1, t); r, s, t ∈ IR}.

QUESTION 5 (a) Use Gauss-Jordan elimination to solve the following system of linear equations: x + 4y − z = 12 3x + 8y − 2z = 4

x + 2y − z = 2 2x + 5y − 3z = 1 x + 4y − 3z = 3

−2x + 3y + 3z = −9 3x − 4y + z = 1 −5x + 7y + 2z = −14 (b) Given the following linear system, determine the value(s) of t so that the system is (i)consistent, (ii) inconsistent. 2x − y = 5 4x − 2y = t (i)t = 10, (ii)t 6= 10 (c) Determine the value(s) of t such that the following linear system has (i) no solution, (ii) one solution and (iii) infinitely many solutions. x + ty − z = 1 −x + (t − 2)y + z = −1 2x + 2y + (a − 2)z = 1 (i)t = 0, (ii)t 6= 0 and t 6= 1, (iii)t=1...