Question

Matrices A and B are square matrices of the same size. Prove Tr(c(A + B)) = C (Tr(A) + Tr(B)).

Answer 1

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))=
`C\cdot` Sum of diagonal elements of A

C(Tr(B))=
`C\cdot` Sum of diagonal elements of B

`C(A+B)=C\cdot (A+B)`

Tr(C(A+B)=Sum of diagonal elements of (C(A+B))

Suppose ,A=
`\left[\begin{array}{ccc}1&0\\1&1\end{array}\right]`

B=
`\left[\begin{array}{ccc}1&1\\1&1\end{array}\right]`

Tr(A)=1+1=2

Tr(B)=1+1=2

C(Tr(A)+Tr(B))=C(2+2)=4C

A+B=
`\left[\begin{array}{ccc}1&0\\1&1\end{array}\right]+\left[\begin{array}{ccc}1&1\\1&1\end{array}\right]`

A+B=
`\left[\begin{array}{ccc}2&1\\2&2\end{array}\right]`

C(A+B)=
`\left[\begin{array}{ccc}2C&C\\2C&2C\end{array}\right]`

Tr(C(A+B))=2C+2C=4C

Hence, Tr(C(A+B)=C(Tr(A)+Tr(B))

Hence, proved.