Answer with Step-by-step explanation:
We are given that two matrices A and B are square matrices of the same size.
We have to prove that
Tr(C(A+B)=C(Tr(A)+Tr(B))
Where C is constant
We know that tr A=Sum of diagonal elements of A
Therefore,
Tr(A)=Sum of diagonal elements of A
Tr(B)=Sum of diagonal elements of B
C(Tr(A))=[tex]C\cdot[/tex] Sum of diagonal elements of A
C(Tr(B))=[tex]C\cdot[/tex] Sum of diagonal elements of B
[tex]C(A+B)=C\cdot (A+B)[/tex]
Tr(C(A+B)=Sum of diagonal elements of (C(A+B))
Suppose ,A=[tex]\left[\begin{array}{ccc}1&0\\1&1\end{array}\right][/tex]
B=[tex]\left[\begin{array}{ccc}1&1\\1&1\end{array}\right][/tex]
Tr(A)=1+1=2
Tr(B)=1+1=2
C(Tr(A)+Tr(B))=C(2+2)=4C
A+B=[tex]\left[\begin{array}{ccc}1&0\\1&1\end{array}\right]+\left[\begin{array}{ccc}1&1\\1&1\end{array}\right][/tex]
A+B=[tex]\left[\begin{array}{ccc}2&1\\2&2\end{array}\right][/tex]
C(A+B)=[tex]\left[\begin{array}{ccc}2C&C\\2C&2C\end{array}\right][/tex]
Tr(C(A+B))=2C+2C=4C
Hence, Tr(C(A+B)=C(Tr(A)+Tr(B))
Hence, proved.
