# Prove that if A and B are similar n x n matrices, then tr(A) = tr(B).

Question
Matrices
Prove that if A and B are similar n x n matrices, then tr(A) = tr(B).

2020-11-12
Step 1
Given that A and B are similar n x n matrices.
Then there exist a non-singular n x n matrix P such that $$B=P^{-1}AP \text{ and } A=PBP^{-1}$$
Step 2
Now
tr(B)=tr(P^{-1}AP)\)
$$= tr(P^{-1}PA) \left[ \because \text{If A,B and C are any matrices } A(B \cdot C)=(AB)C , \text{ provided both sides are defined} \right]$$
$$=tr(IA) , \text{ where I is the identity matrix}$$
$$=tr(A) \left[ \because \text{ for any } n \times n \text{matrix} A,I \cdot A =A \right] ### Relevant Questions asked 2020-11-12 Prove that if A and B are n x n matrices, then tr(AB) = tr(BA). asked 2021-03-02 Zero Divisors If a and b are real or complex numbers such thal ab = O. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n x n matrices such that AB = 0. is il true lhat we must have A = 0 or B = 0? Prove lhe resull or find a counterexample. asked 2021-03-18 Let A and B be similar nxn matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if \(X^2=X$$ )
Let A and B be $$n \times n$$ matrices. Recall that the trace of A , written tr(A),equal
$$\sum_{i=1}^nA_{ii}$$
Prove that tr(AB)=tr(BA) and $$tr(A)=tr(A^t)$$
Suppose that A and B are diagonalizable matrices. Prove or disprove that A is similar to B if and only if A and B are unitarily equivalent.
Prove: If A and B are $$n \times n$$ diagonal matrices, then
AB = BA.
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
Let $$A=I_2$$ and $$B=\begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix}$$ . Discuss the validity of the following statement and cite reasons for your conclusion:
that $$(BA^{-1})^T(A^{-1}B^T)^{-1} = I$$
If A and B are $$n \times n$$ diagonalizable matrices , then A+B is also diagonalizable.