Tutorial Sheet 2
II B. Tech. (Common)
Mathematics-II
(MCI102)
1. Find the eigenvalues and



1 −1 2
1



(a) 0 1 0
(b) 2
3
1 2 1
eigenvectors of the following matrices:



3 2 1
0
0 0
0 1 0
1 

1 0 (c) 
0 2 −1 0 
2 0
0 0 0 1/2


1 2 0
2. Verify the Caley-Hamilton theorem for the matrix A = −1 1 2.
1 2 1
Also (a) obtain A−1 and A3 , (b) find the eigenvalues of A and A2 and verify that the eigenvalues of A2
are squares of those of A, (c) find the spectral radius of A.


3 −1 1
1 2
3. By using the Cayley-Hamilton theroem, find A8 for A =
and find A4 for A = −1 5 −1
2 −1
1 −1 3


2 −1 1
4. Find the characteristic equation of the matrix A = −1 2 −1 and then find A−1 by using the
1 −2 2
6
Cayley-Hamilton theorem. Also, find the value of A − 6A5 + 9A4 − 2A3 − 12A2 + 23A − 9I.
5. Examine whether A is similar to B, where
5 5
1 2
(a) A =
and B =
.
−2 0
−3 4
1 0
1 1
(b) A =
and B =
.
0 1
0 1


3 1 −1
6. Show that the matrix A = −2 1 2  is diagonalizable. Then find a matrix P such that P −1 AP is a
0 1 2
diagonal matrix. Then obtain the matrix B = A2 + 5A + 3I.
7. Examine whether the matrix A is diagonalizable, where A is given by




1 2 2
−2 2 −3
(a) A =  0 2 1 (b) A =  2
1 −6
−1 2 2
−1 −2 0
If A is diagonalizable, then find a matrix P such that P −1 AP is a diagonal matrix.
T T
8. The eigenvectors of a 3×3 matrix A corresponding to the eigenvalues 1, 1, 3 are 1 0 −1 , 0 1 −1 ,
T
1 1 0 respectively. Find the matrix A.
9. Obtain the symmetric matrix for the quadratic form:
(a) 2x21 + 3x1 x2 + x22 (b) x21 + 2x1 x2 − 4x1 x3 + 6x2 x3 − 5x22 + 4x23
10. Reduce the following quadratic form to the canonical form by an orthogonal transformation. Also, specify
the matrix of transformation in each case.
(a) 3x21 + 3x22 + 3x23 + 2x1 x2 + 2x1 x3 − 2x2 x3 (b) x2 + 2y 2 + 2z 2 − 2xy − 2yz + zx
11. Determine the nature, index, and signature of the following:
(a) x2 + 2y 2 + 2z 2 − 2xy − 2yz + zx (b) 2x1 x2 + 2x1 x3 + 2x2 x3
12. Solve the following differential equations (initial value problems) by matrix method:
(a) x′′ − 2x′ − 3x = 0, x(0) = 0, x′ (0) = 1 (b) y ′′ + µ2 y = 0, y(0) = 1, y ′ (0) = µ
dx2
1
(c) dx
dt = x1 + x2 , dt = lx1 + x2 , x1 (0) = 10, x2 (0) = 70
1
13. Show that:
(a) The eigenvalues of a Hermitian matrix are real.
(b) The eigenvalues of a skew-Hermitian matrix are zero or pure imaginary.
(c) The
 eigenvalues of a unitary matrix are of magnitude 1.
2 −2 −4
1 1
(d) −3 3
4  is an idempotent matrix. Also, show that
is involutory.
0 −1
1 −2 −2
(e) Every skew-Hermitian matrix A can be expressed as B + iC, where B is real skew-symmetric matrix
and C is real symmetric matrix.
(f) λ = 0 is an eigenvalue of a matrix A if and only if A is a singular matrix.


0 2b c
14. Determine the values of a, b, and c if the matrix a b −c is orthogonal.
a −b c
2