Department of Physics St. Paul’s C. M. College Page 1

Problem Set – Matrices

1. Consider Pauli’s spin matrices: 𝜎𝑥 = 0 11 0

, 𝜎𝑦 = 0 −𝑖𝑖 0

, 𝜎𝑧 = 1 00 −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 = 𝜎𝑦

2 = 𝜎𝑧2 = 𝐼, 𝐼 is a second order unit matrix.

(d) Show that 𝜎2 ,𝜎𝑥 = 𝜎2 ,𝜎𝑦 = 𝜎2 ,𝜎𝑧 = 0, where 𝜎2 = 𝜎𝑥2 + 𝜎𝑦

2 + 𝜎𝑧2.

(e) Are these matrices orthogonal?

(f) Are these matrices Hermitian?

(g) Are these matrices unitary?

2. Show that for any matrix 𝐻, 𝐻 + 𝐻† and 𝑖 𝐻 − 𝐻† are 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. Given 𝐴 and 𝐵 both are symmetric matrices. Show that the commutator of 𝐴 and 𝐵, that is,

𝐴𝐵 – 𝐵𝐴 is skew-symmetric.

5. Given A and B both are Hermitian matrices. Show that the commutator of A and B, that is, AB – BA is

skew-Hermitian.

6. If 𝐴 and 𝐵 are two Hermitian matrices, prove that 𝐴𝐵 is Hermitian only if 𝐴 and 𝐵 commute.

7. Show that the determinant of an orthogonal matrix is ±1.

8. Consider a third order square matrix 𝐴 = 2 1 01 0 10 2 1

. Obtain the cofactor matrix and hence the adjoint

matrix of 𝐴. Also find the inverse of 𝐴, i.e. 𝐴−1. Is 𝐴 self-adjoint matrix?

9. Show that (𝐴𝐵)−1 = 𝐵−1𝐴−1, where 𝐴 and 𝐵 are two square matrices of same order.

10. Obtain the eigen values and normalized eigen vectors of matrix 𝐴 = 1 2 0

2 0 00 0 0

.

11. Obtain the trace and determinant of the matrix 𝐴 = 1 00 −1

. Hence obtain its eigen values. Also find

the normalized eigen vectors of 𝐴.

12. Prove that at least one eigen value of a singular matrix is zero.

13. Prove that eigen values of a unitary matrix are of unit magnitude.

14. For a nonsingular matrix 𝐴, prove that the eigen values of its inverse matrix are reciprocal of the eigen

values of the original matrix 𝐴.

Department of Physics St. Paul’s C. M. College Page 2

15. Show that the eigen values of a hermitian matrix are real and the eigen vectors corresponding to

different eigen values are orthogonal.

16. Eigen values of antihermitian matrix are either purely imaginary or zero.

17. Show that for a diagonal matrix the eigen values are equal to its elements along the principal diagonal.

18. Show that if 𝜆 is an eigen value of a matrix 𝐴, the eigen value of the matrix 𝐴𝑛 (𝑛 is an integer) is 𝜆𝑛 .

19. Show that if 𝜆 is an eigen value of a matrix 𝐴, the eigen value of the matrix 𝑒𝐴 is 𝑒𝜆.

20. Diagonalise the matrix 𝐴 = 1 2 02 −1 00 0 1

. Also find the diagonalising matrix. Show that the matrix A

satisfies Caley – Hamilton’s theorem.

21. 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.

22. Given 𝐴 = 1 −1 12 −1 01 0 0

. Find 𝐴2 and show that 𝐴2 = 𝐴−1.

Review of CU Exam. Papers:

CU – 2015

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

2. Find the eigenvalues and normalized eigen vectors of the matrix cos 𝜃 sin𝜃− sin𝜃 cos𝜃

. [4]

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]

2. Find the eigen values and normalized eigen vectors of the matrix 𝐴 = 1 14 1

. [2+3]

3. If a matrix is both Hermitian and unitary, show that all its eigen values are ±1. [3]

4. Show that the product of two symmetric matrices is symmetric if they commute. [2]

CU – 2013

1. Prove that simultaneous eigen vectors exists for two matrices if they commute. [2]

2. Find the eigen values and normalized eigen vectors of the matrix 𝐴 = 5 22 2

. [4]

3. Consider the matrices 𝜎𝑥 = 0 11 0

and 𝜎𝑦 = 0 −𝑖𝑖 0

. Which of the matrices is/are Hermitian?

Which of the matrices is/are unitary? Find 𝜎𝑥𝜎𝑦 − 𝜎𝑦𝜎𝑥 . [2+2+2]

Department of Physics St. Paul’s C. M. College Page 3

CU – 2012

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

2. Consider the following transformation in three dimensions:

𝑥′ = 𝑥 cos𝜃 + 𝑦 sin𝜃 , 𝑦′ = −𝑥 sin𝜃 + 𝑦 cos𝜃, 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 eigen values of a Hermitian matrix are real. [2]

4. Find the eigen values of the matrix 𝐴 = 1 11 1

. [3]

5. Prove 𝑒𝑖𝜃𝜎 = cos𝜃 1 + 𝑖 sin𝜃 𝜎 where 𝜎 is a matrix with 𝜎2 = 1 and 𝜃 is a real quantity. [2]

CU – 2011

1. If 𝑆 and 𝐴 are unitary matrices, show that 𝑆−1𝐴𝑆 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]

2. Find the eigen values and normalized eigen vectors of the matrix 𝐴 = 0 −𝑖𝑖 0

. [2+2]

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 𝐴𝑇𝐴 = 𝐼. [2]

2. Determine the sum and product of the eigen values of the matrix 𝑎 𝑏𝑐 𝑑

. [2]

3. Find the eigen values and eigen vectors of the matrix 𝐴 = 1 −2−2 −2

. [4]