DSC-7: Mathematical Physics II

Problem Set – 1 (Matrices)

0 1 0 −𝑖 1 0
1. Consider Pauli’s spin matrices: 𝜎𝑥 = ( ), 𝜎𝑦 = ( ), 𝜎𝑧 = ( ).
1 0 𝑖 0 0 −1
(a) Show that [𝜎𝑥 , 𝜎𝑦 ] = 2𝑖𝜎𝑧 , [𝜎𝑦 , 𝜎𝑧 ] = 2𝑖𝜎𝑥 , [𝜎𝑧 , 𝜎𝑥 ] = 2𝑖𝜎𝑦 . Given, [𝐴, 𝐵] = 𝐴𝐵 − 𝐵𝐴.
(b) Show that {𝜎𝑥 , 𝜎𝑦 } = {𝜎𝑦 , 𝜎𝑧 } = {𝜎𝑧 , 𝜎𝑥 } = 0. Given {𝐴, 𝐵} = 𝐴𝐵 + 𝐵𝐴.
(c) Show that 𝜎𝑥2 = 𝜎𝑦2 = 𝜎𝑧2 = 𝐼, 𝐼 is a second order unit matrix.
(d) Show that [𝜎 2 , 𝜎𝑥 ] = [𝜎 2 , 𝜎𝑦 ] = [𝜎 2 , 𝜎𝑧 ] = 0, where 𝜎 2 = 𝜎𝑥2 + 𝜎𝑦2 + 𝜎𝑧2 .
(e) Are these matrices orthogonal?
(f) Are these matrices Hermitian?
(g) Are these matrices unitary?
2. Show that every square matrix can be uniquely represented as the sum of a symmetric and a skew-symmetric
matrix.
3. Write down the following matrix as the sum of a symmetric and a skew-symmetric matrix.
1 2 4
𝐴 = (−2 5 3)
−1 6 3
2. Show that for any matrix 𝐻, 𝐻 + 𝐻 † and 𝑖(𝐻 − 𝐻 †), 𝐻𝐻 † are all Hermitian but (𝐻 − 𝐻 †) is anti-Hermitian.
3. Show that every square matrix can be uniquely represented as the sum of a Hermitian and an anti-Hermitian
matrix.
4. Represent the following matrix 𝐴 as the sum of:
3 + 2𝑖 1−𝑖 5
𝐴 = (−2 + 7𝑖 6𝑖 −1 + 4𝑖 )
4 − 3𝑖 2 + 2𝑖 6+𝑖
(i) a real and a purely imaginary matrix.
(ii) a symmetric and a skew-symmetric matrix.
(iii) a Hermitian and a skew-Hermitian matrix.
5. Given 𝐴 and 𝐵 both are symmetric matrices. Show that the commutator of 𝐴 and 𝐵, that is, 𝐴𝐵 – 𝐵𝐴 is skew-
symmetric.
6. Given A and B both are Hermitian matrices. Show that the commutator of A and B, that is, AB – BA is skew-
Hermitian.
7. If 𝐴 and 𝐵 are Hermitian matrices, show that (𝐴𝐵 + 𝐵𝐴) and 𝑖(𝐴𝐵 − 𝐵𝐴) are also Hermitian.
8. If 𝐴 and 𝐵 are two Hermitian matrices, prove that 𝐴𝐵 is Hermitian only if 𝐴 and 𝐵 commute.
9. If 𝐻 is a Hermitian matrix, show that 𝑒 𝑖𝐻 is unitary matrix.
10. Show that the determinant of an orthogonal matrix is ±1.
cos 𝜃 sin 𝜃
11. Find the inverse of the matrix 𝐴 = ( )
sin 𝜃 cos 𝜃

© Dr. P. Mandal Page 1

M: 8902442561
 2 1 0
12. Consider a third order square matrix 𝐴 = (1 0 1). Obtain the cofactor matrix and hence the adjoint
0 2 1
matrix of 𝐴. Also find the inverse of 𝐴, i.e. 𝐴−1 . Is 𝐴 self-adjoint matrix?
13. ‘Every non-singular square matrix has a unique inverse’ – Prove.
14. Show that (𝐴𝐵)−1 = 𝐵 −1 𝐴−1 , where 𝐴 and 𝐵 are two square matrices of same order.
15. Find the matrix 𝐵 such that 𝐴 = 𝐵𝐶 where
2 3 −2 1 2 −1
𝐴 = (4 −1 −2) , 𝐶=(2 −1 −1)
0 1 0 −1 2 0
1 √2 0
16. Obtain the eigenvalues and normalized eigenvectors of matrix 𝐴 = (√2 0 0).
0 0 0
1 0
17. Obtain the trace and determinant of the matrix 𝐴 = ( ). Hence obtain its eigenvalues. Hence obtain
0 −1
the eigenvalues of 𝐴−1 . Also find the normalized eigenvectors of 𝐴.
18. Prove that at least one eigenvalue of a singular matrix is zero.
19. Prove that eigenvalues of a unitary matrix are of unit magnitude.
20. For a nonsingular matrix 𝐴, prove that the eigenvalues of its inverse matrix are reciprocal of the eigenvalues
of the original matrix 𝐴.
21. Show that the eigenvalues of a Hermitian matrix are real and the eigenvectors corresponding to different
eigenvalues are orthogonal.
22. Eigenvalues of skew-Hermitian matrices are either purely imaginary or zero.
23. Show that for a diagonal matrix the eigenvalues are equal to its elements along the principal diagonal.
24. Prove that eigenvalues are invariant under a similarity transformation.
25. Show that if 𝜆 is an eigenvalue of a matrix 𝐴 corresponding to an eigenvector 𝑋, the eigenvalue of the matrix
𝐴𝑛 (𝑛 is an integer) is 𝜆𝑛 corresponding to the same eigenvector 𝑋.
26. Show that if 𝜆 is an eigenvalue of a matrix 𝐴 corresponding to an eigenvector 𝑋, the eigenvalue of the matrix
𝑒 𝐴 is 𝑒𝜆 corresponding to an eigenvector 𝑋.
27. If 𝑌 is an eigenvector of 𝐵 = 𝑅 −1 𝐴𝑅 corresponding to an eigenvalue 𝜆, then 𝑈 = 𝑅𝑌 is an eigenvector of 𝐴
corresponding to the same eigenvalue.
1 2 0
28. Diagonalize the matrix 𝐴 = (2 −1 0). Also find the diagonalising matrix. Show that the matrix A
0 0 1
satisfies Caley – Hamilton’s theorem.
29. If a matrix 𝐴 satisfies a relation 𝐴2 + 𝐴 − 𝐼 = 0, prove that 𝐴−1 exists and that 𝐴−1 = 𝐴 + 𝐼, where 𝐼 is a
unit matrix of same order as 𝐴.
1 −1 1
30. Given 𝐴 = (2 −1 0). Find 𝐴2 and show that 𝐴2 = 𝐴−1 .
1 0 0

© Dr. P. Mandal Page 2

M: 8902442561
31. Verify Cayley-Hamilton principle for the following matrix 𝐴 and hence find out 𝐴−1 :
1 2
𝐴=( )
2 −1
31. Solve the following differential equations using the method of matrices:
𝑑𝑥 𝑑𝑦
= 𝑥 + 𝑦, = 4𝑥 + 𝑦
𝑑𝑡 𝑑𝑡

Review of CU Exam. Papers:

CU – 2022 (CBCS)

1. Prove that the eigenvalue of a skew Hermitian matrix is purely imaginary. [2]

3 𝑖
2. Find the eigenvalues of the matrix ( ). [2]
−𝑖 3

3. The matrices 𝐴 and 𝐵 satisfy (𝐴𝐵)𝑇 + 𝐵 −1 𝐴 = 0. Prove that if 𝐵 is orthogonal, then 𝐴 is anti- symmetric.

1 𝑎
4. Find out the eigenvalues and normalized eigenvectors of the matrix 𝑀 = ( ) (𝑎 ≠ 0). Find out 𝑀𝑛 where
0 1
𝑛 is a positive integer. [1+2+2]

1 0 1
5. Explain whether the inverse of the following matrix exists. (2 2 0). [3]
1 1 0

6. Show that for an orthogonal matrix, each column is orthogonal to other ones. [3]

7. Show that two similar matrices have the same characteristic polynomial. [3]

8. If 𝐴2 = 𝐴, then show that 𝑒 𝜃𝐴 = 𝕀 + (𝑒 𝜃 − 1)𝐴. [2]

0 𝑎 𝑏
9. Given 𝐴 = (−𝑎 0 𝑐 ) what can you comment on the nature of eigenvalues of 𝐴 without solving the
−𝑏 −𝑐 0
characteristic equation. [2]

CU – 2021 (CBCS)

1. Given a unitary matrix 𝑈, show that 𝑈 −1 𝐻𝑈 is Hermitian if 𝐻 is a Hermitian matrix. [2]

𝑥1
2. Find a symmetric matrix 𝑆 such that 𝑄 = 𝑋𝑇 𝑆𝑋 where 𝑄 = 𝑥12 + 2𝑥1 𝑥2 − 3𝑥22 and 𝑋 = (𝑥 ). [2]
2

3. Show that the eigenvalues 𝜆 of a two-dimensional invertible real-valued matrix 𝐴 obeying 𝐴−1 = 𝐴†, satisfy
|𝜆|2 = 1. [3]
−1 𝐴𝐵
4. Show that if 𝐵 is an invertible matrix, then 𝐵 −1 𝑒 𝐴 𝐵 = 𝑒 𝐵 . [3]

5. Solve the system of equations by Matrix method

© Dr. P. Mandal Page 3

M: 8902442561
 𝑑𝑦 𝑑𝑧
= 𝑧, = −𝑦
𝑑𝑡 𝑑𝑡

with initial conditions 𝑦(0) = 1, 𝑦̇(0) = 0. [4]

6. Let a unitary matrix 𝑈 can be written as 𝑈 = 𝐴 + 𝑖𝐵, where 𝐴 and 𝐵 are Hermitian matrices having non-
degenerate eigenvalues. Show that 𝐴2 + 𝐵2 = 𝐼. [3]

𝑎 𝑏 1
7. Show that for a 2 × 2 matrix 𝐴 = ( ), det 𝐴 = [(𝑇𝑟 𝐴)2 − 𝑇𝑟(𝐴2 )], where 𝑇𝑟 represents trace. [3]
𝑐 𝑑 2

𝑎 ℎ cos 𝜃 sin 𝜃
8. If a matrix 𝐴 = ( ) is transformed to the diagonal form 𝐵 = 𝑈𝐴𝑈 −1 where 𝑈 = ( ), show
ℎ 𝑏 −sin 𝜃 cos 𝜃
1 2ℎ
that 𝜃 = tan−1 ( ). [4]
2 𝑎−𝑏

CU – 2019 (CBCS)

1. If 𝑀 is an orthogonal matrix, prove that det 𝑀 = ±1. [2]

2. Show that any matrix can be written as the sum of a Hermitian matrix and a skew-Hermitian matrix. [2]
3. Prove that the eigenvalues of a Hermitian matrix are real. [2]
4. Prove that the product of two orthogonal matrices is orthogonal. [2]
1 2
5. Find the eigenvalues and normalized eigenvectors of the matrix ( ). [3]
4 3
6. State Cayley-Hamilton’s theorem. A 3 × 3 matrix 𝑀 satisfies the equation 𝑀2 − 3𝑀 + 2𝐼 = 𝑂, where 𝐼 is the
identity matrix. Find det(𝑀), given 𝑇𝑟𝑎𝑐𝑒(𝑀) = 6. [1+3]
7. If two matrices 𝐴 and 𝐵 satisfy 𝐴𝐵 = 𝑂 and 𝐴 is non-singular, prove that 𝐵 = 𝑂. [2]
8. Prove that for a diagonalizable matrix 𝐴, exp 𝑇𝑟𝑎𝑐𝑒 𝐴 = det exp 𝐴. [4]

CU – 2018 (CBCS)

1. Show that a Hermitian matrix remains Hermitian under a unitary transformation. [2]
2 4
2. Find the eigenvalues and eigenvectors of the matrix 𝑀 = ( ). Are the two eigenvectors orthogonal?
1 2
[4+1]
3. If a matrix 𝑀 satisfies 𝑀2 = 1, what are its possible eigenvalues? [2]
4. Show that Trace (𝐴𝐵) = Trace (𝐵𝐴) where 𝐴 and 𝐵 are two 𝑛 × 𝑛 matrices. [3]
5. Show that the determinant of a unitary matrix is a complex number with modulus unity. [3]
4. Solve the system of equations:
𝑑𝑥
=𝑥+𝑦
𝑑𝑡
𝑑𝑦
= 3𝑥 − 𝑦
𝑑𝑡
with initial conditions 𝑥(𝑡 = 0) = 0, 𝑦(𝑡 = 0) = 1, by matrix method. [4]

© Dr. P. Mandal Page 4

M: 8902442561
CU – 2018

1. Show that the sum of the eigenvalues of a matrix is equal to the trace of the matrix. [2]
2 3−𝑖
2. Find the eigenvalues and normalized eigenvectors of the matrix ( ). [4]
3+𝑖 −1
3. The matrices A and B satisfy the equation (𝐴𝐵)𝑇 + 𝐵 −1 𝐴 = 0. Prove that if B is orthogonal, then A is
antisymmetric. [2]
4. Prove that Hermitian matrices have real eigenvalues and mutually orthogonal eigenvectors. [2+2]

CU – 2017

1. Prove that the product of two Hermitian matrices 𝐴 and 𝐵 is Hermitian matrices 𝐴 and 𝐵 is Hermitian only if
𝐴 and 𝐵 commute. [2]
2. Show that any square matrix can be written as the sum of a symmetric and antisymmetric matrix. [2]
2 2
3. Find the eigenvalues and the normalized eigenvectors of the matrix 𝑀 = ( ). [2+3]
2 −1
4. A normal matrix N is defined by the relation 𝑁𝑁 † = 𝑁 †𝑁. If N is written as (𝐴 + 𝑖𝐵), where 𝐴 and 𝐵 are
Hermitian, show that A and B commute. [3]

CU – 2016

1. If 𝐻 is a Hermitian matrix and 𝑈 is a unitary matrix, show that 𝑈 −1 𝐻𝑈 is Hermitian. [2]

2. If a matrix 𝐴 has eigenvalue 𝜆 with respect to an eigenvector 𝑌, then show that 𝑒 𝐴 has eigenvalue 𝑒𝜆 with
respect to the same eigenvector 𝑌. [3]
1 1
3. Find out the eigenvalues and normalized orthogonal eigenvectors of the matrix ( ). [4]
0 1
0 1 0 −𝑖 1 0
4. Consider the matrices 𝜎1 = ( ), 𝜎2 = ( ), 𝜎3 = ( ).
1 0 𝑖 0 0 −1
Which of the matrices is/are Hermitian? Which is/are unitary? [1+2]

CU – 2015

1. If 𝑀 is an orthogonal matrix, prove that Det 𝑀 = ±1. [2]

2. Find the eigenvalues and normalized eigenvectors of the matrix (

cos 𝜃 sin 𝜃 ). [4]
− sin 𝜃 cos 𝜃
3. Prove that all eigenvalues of a Hermitian matrix are real. [2]

CU – 2014

1. Show that a Hermitian matrix remains Hermitian under a unitary transformation. [2]
1 1
2. Find the eigenvalues and normalized eigenvectors of the matrix 𝐴 = ( ) . [2+3]
4 1
3. If a matrix is both Hermitian and unitary, show that all its eigenvalues are ±1. [3]
4. Show that the product of two symmetric matrices is symmetric if they commute. [2]

© Dr. P. Mandal Page 5

M: 8902442561
CU – 2013
1. Prove that simultaneous eigenvectors exists for two matrices if they commute. [2]
5 2
2. Find the eigenvalues and normalized eigenvectors of the matrix 𝐴 = ( ) . [4]
2 2
0 1 0 −𝑖
3. Consider the matrices 𝜎𝑥 = ( ) and 𝜎𝑦 = ( ). Which of the matrices is/are Hermitian? Which of
1 0 𝑖 0
the matrices is/are unitary? Find 𝜎𝑥 𝜎𝑦 − 𝜎𝑦 𝜎𝑥 . [2+2+2]

CU – 2012
1. Given a unitary matrix 𝑈, show that 𝑈 −1 𝐴𝑈 is Hermitian if 𝐴 is Hermitian. [2]
2. Consider the following transformation in three dimensions:
𝑥 ′ = 𝑥 cos 𝜃 + 𝑦 sin 𝜃 , 𝑦 ′ = −𝑥 sin 𝜃 + 𝑦 cos 𝜃, 𝑧 ′ = 𝑧.
(i) Write down the transformation matrix 𝐴(𝜃).
(ii) Show that 𝐴(𝜃1 )𝐴(𝜃2 ) = 𝐴(𝜃1 + 𝜃2 ).
(iii) Is 𝐴(𝜃) unitary? [1+2+2]
3. Show that all the eigenvalues of a Hermitian matrix are real. [2]
1 1
4. Find the eigenvalues of the matrix 𝐴 = ( ) . [3]
1 1
5. Prove 𝑒𝑖𝜃𝜎 = cos 𝜃 1 + 𝑖 sin 𝜃 𝜎 where 𝜎 is a matrix with 𝜎 2 = 1 and 𝜃 is a real quantity. [2]

CU – 2011
1. If 𝑆 and 𝐴 are unitary matrices, show that 𝑆 −1 𝐴𝑆 is also a unitary matrix. [2]
2. Give an example where the product of two matrices is a null matrix, but none of them is null matrix. [2]
3. For any matrix 𝑋, if 𝑋𝐴 = 𝐴𝑋 = 𝐴 for every 𝐴, then show that 𝑋 = 𝐼 where 𝐼 is the identity matrix. [3]
4. If 𝑋 and 𝑌 are Hermitian matrices, show that 𝑖(𝑋𝑌 − 𝑌𝑋) is Hermitian. [2]
5. Show that for a square matrix 𝐴, Tr(𝐴) = sum of eigen values and det 𝐴 = product of eigen values. [3]

CU – 2010
1. Show that a Hermitian matrix remains Hermitian under unitary transformation. [2]
0 −𝑖
2. Find the eigenvalues and normalized eigenvectors of the matrix 𝐴 = ( ) . [2+2]
𝑖 0

CU – 2009
1. Let 𝐴 be a square finite dimensional matrix with real entries such that 𝐴𝐴𝑇 = 𝐼, where 𝐴𝑇 denotes the
transpose of 𝐴. Show that 𝐴𝑇 𝐴 = 𝐼. [2]
𝑎 𝑏
2. Determine the sum and product of the eigenvalues of the matrix ( ). [2]
𝑐 𝑑
1 −2
3. Find the eigenvalues and eigenvectors of the matrix 𝐴 = ( ) . [4]
−2 −2

© Dr. P. Mandal Page 6

M: 8902442561