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The purpose of this paper is to discuss the theorems for the trace of any positive integer power of 2 × 2 real matrix. We obtain a new formula to compute trace of any positive integer power of 2 × 2 real matrix
A, in the terms of Trace of
A (Tr
A) and Determinant of
A (Det
A), which are based on definition of trace of matrix and multiplication of the matrixn times, where
n is positive integer and this formula gives some corollary for Tr
A^{n} when Tr
A or Det
A are zero.

Traces of powers of matrices arise in several fields of mathematics, more specifically, Network Analysis, Numbertheory, Dynamical systems, Matrix theory, and Differential equations [^{3})/6, where A is the adjacency matrix of the graph [

for all integer matrices A, all primes p, and all r ∊ Z. The invariants of dynamical systems are described in terms of the traces of powers of integer matrices, for example in studying the Lefschetz numbers [^{n} and A^{−n}, n ∊ Z, is proposed in [

Trace of a

The computation of the trace of matrix powers has received much attention. In [^{q} is also symmetric positive definite, and it holds [

This formula is restricted to the matrix A. Also we have other formulae [

But for many cases, this formula is time consuming. For example

Consider a matrix ^{5}. Eigenvalues of A are

Computation of this value is time consuming. Therefore, other formulae to compute trace of matrix power are needed. Now we give new theorems and corollaries to compute trace of matrix power. Our estimation for the trace of A^{n} is based on the multiplication of matrix.

Theorem 1. For even positive integer n and 2 × 2 real matrix A,

Proof. Consider a matrix

Then

and

Now

Then

Now again

Then

Now replace A by A^{2} in (2.3), we have

Again replace A by A^{2} in (2.5), we have

Now again replace A by A^{2} in (2.6), we have

Now we observe from (2.3), (2.6), (2.7) and (2.8) that

Continuing this process up to n terms we get

Finally from above, we get

Hence the proof is completed.

Theorem 2. For odd positive integer n and 2 × 2 real matrix A,

Proof. Consider a matrix A as in theorem 1, we have from (1.4) and (1.6).

Now we observe from (2.5) and (2.11) that

Now we continuing this as in Theorem 1, we get TrA^{n} same as Theorem 1. But here r varies up to

Corollary 1: For any positive integer n and 2 × 2 real singular matrix A,

Proof: For singular matrix A, DetA = 0. Hence proof follows from Theorem 1 and Theorem 2.

Corollary 2: For 2 × 2 real matrix A with TrA = 0.

1)

2)

Proof. Proof follows from theorem 1 and theorem 2.

Corollary 3: For 2 × 2 real matrix A with TrA = 0 and DetA = 0.

Proof. Proof follows from Corollary 2.

Example 1. Consider a matrix ^{5}.

Here

Example 2. Consider a matrix ^{10}.

Here

Example 3. Consider a matrix ^{2015}.

Here TrA = 0, DetA = −2 and n = 2015, which is odd, hence by corollary 2, we get TrA^{2015} = 0.

Example 4. Consider a matrix ^{100}.

Here A is a singular matrix with Trace 1, and then by Corollary 1, we have

After to discuss Theorems 1 and 2, Corollaries 1, 2 and 3, we are able to find trace of any integer power of a 2 × 2 real matrix. In future, we can be developed similar results for 3 × 3 real matrices.

We would like to hardly thankful with great attitude to Director, Maulana Azad National Institute of Technology, Bhopal for financial support and we also thankful to HOD, Department of Mathematics of this institute for giving me opportunity to expose my research in scientific world.

Jagdish Pahade,Manoj Jha, (2015) Trace of Positive Integer Power of Real 2 × 2 Matrices. Advances in Linear Algebra & Matrix Theory,05,150-155. doi: 10.4236/alamt.2015.54015