Calculate the trace of a square matrix using the matrix trace calculator. Select the dimension of a matrix among 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8 and fill the matrix with values. The calculator will calculate the trace of the matrix for you.

The trace of a matrix is the sum of all the elements in its main diagonal.

Only a square matrix has a trace.

For example, there is a matrix A, The trace of a matrix A is denoted by tr(A).

Trace of a matrix is not just a mathematical concept but it has real world applications too.

Its used in following fields:

1. Quantum Mechanics

2. Computer Graphics

3. Machine Learning

4. Robotics

5. Physics

6. Economics

Say, there is a matrix A

1. tr(A) = tr(A^T)

Trace of a matrix is always equal to the trace of its transpose matrix

2. tr(A + B) = tr(A) + tr(B)

The trace of the sum of two matrices is equal to the sum of the traces of the matrices

3. tr(λA) = λ * tr(A)

The trace of a scalar multiple of a matrix is equal to the scalar multiple of the trace of the matrix

4. tr(AB) = tr(BA)

This is the most interesting property of trace of a matrix. The order of multiplication of two matrices doesn't matter. The trace of the resultant matrix is always the same value

And the multiplication property holds true for any number of matrices. tr(PQR) = tr(RPQ) = tr(QRP) = tr(PQR) = tr(RQP) = tr(QPR) = ... = tr(...)

To calculate the trace of a matrix, the matrix must be a square matrix.

Take matrix A. Matrix A is a 2x2 squar matrix.

The trace of matrix A is calculated as follows:

A = [ [a11, a12], [a21, a22] ]

tr(A) = a11 + a22

The trace of matrix A = a11 + a22

Take matrix M. Matrix M is a 3x3 square matrix.

The trace of matrix M is calculated as follows:

M = [ [a11, a12, a13], [a21, a22, a23], [a31, a32, a33] ]

tr(M) = a11 + a22 + a33

The trace of matrix M = a11 + a22 + a33

Take matrix N. Matrix N is a 4x4 square matrix.

The trace of matrix N is calculated as follows:

N = [ [a11, a12, a13, a14], [a21, a22, a23, a24], [a31, a32, a33, a34], [a41, a42, a43, a44] ]

tr(N) = a11 + a22 + a33 + a44

The trace of matrix N = a11 + a22 + a33 + a44

There is a matrix P which is a 2x2 matrix.

P = [ [2, 3], [4, 5] ]

The trace of matrix P is calculated as follows:

tr(P) = 2 + 5

tr(P) = 7

The calculator above is designed to calculate the trace of a square matrix.

Select the dimension of the matrix you want to calculate the trace for.

Fill the matrix with values.

Click on the calculate button to calculate the trace of the matrix.

You can also fill the matrix with random values by clicking on the random button.

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