Trace of a Matrix

Let A = [a_ij]_mxn be a square matrix. Then, the sum of all diagonal elements of A is called the trace of A and is denoted by tr(A).

Thus, \(tr(A)=\sum\limits_{i=1}^{n}{{{a}_{ii}}={{a}_{11}}+{{a}_{22}}+….+{{a}_{nn}}}\).

Properties of Trace of Matrix:

Let A = [a_ij] and B = [b_ij] are two square matrix of order n then

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

ii. tr (AB) = tr (BA)

iii. tr (KA) = k tr (A), where k is a scalar

iv. tr (A’) = tr (A)

Examples 1: Find the trace of matrix \(\left[ \begin{matrix} 2 & -7 & 9 \\ 0 & 3 & 2 \\ 8 & 9 & 4 \\\end{matrix} \right]\).

Solution: \(A=\left[ \begin{matrix} 2 & -7 & 9 \\ 0 & 3 & 2 \\ 8 & 9 & 4 \\\end{matrix} \right]\),

We can use the formula \(tr(A)=\sum\limits_{i=1}^{n}{{{a}_{ii}}={{a}_{11}}+{{a}_{22}}+….+{{a}_{nn}}}\),

\(tr(A)=\sum\limits_{i=1}^{n}{{{a}_{ii}}={{a}_{11}}+{{a}_{22}}+{{a}_{33}}}\),

a₁₁ = 2; a₂₂ = 3 and a₃₃ = 4

Now tr(A) = 2 + 3 + 4 = 9

Examples 2: Find the trace of matrix \(\left[ \begin{matrix} 2 & 7 & 1 \\ 0 & 2 & 2 \\ 2 & 5 & 4 \\\end{matrix} \right]\).

Solution: \(A=\left[ \begin{matrix} 2 & 7 & 1 \\ 0 & 2 & 2 \\ 2 & 5 & 4 \\\end{matrix} \right]\),

We can use the formula \(tr(A)=\sum\limits_{i=1}^{n}{{{a}_{ii}}={{a}_{11}}+{{a}_{22}}+….+{{a}_{nn}}}\),

\(tr(A)=\sum\limits_{i=1}^{n}{{{a}_{ii}}={{a}_{11}}+{{a}_{22}}+{{a}_{33}}}\),

a₁₁ = 2; a₂₂ = 2 and a₃₃ = 4

Now tr(A) = 2 + 2 + 4 = 8.