# True or False - Determinants of two similar matrices are the same. Explain. Question
Matrices True or False - Determinants of two similar matrices are the same. Explain. 2021-01-14
Step 1
According to the given information, it is required to say whether the given statement is true or not.
The given statement is true as if two matrices are similar then their determinant are equal.
Step 2
Let A and B are similar matrices.
When two matrices are similar then there exists a non-singular matrix such that: let the non singular matrix be L then ,
$$L^{-1}AL=B$$
take determinant both sides
$$det(L^{-1}AL)=det(B)$$
$$det(L^{-1}AL)=det(B)$$
$$det(L^{-1})det(A)det(L)=det(B) \left[\text{ as det }(AB)=det(A)det(B)\right]$$
$$det(L^{-1})det(L)det(A)=det(B) \left[\text{ determinant of any matrix is a number so it is commutative }\right]$$
$$det(L^{-1}L)det(A)=det(B)$$
$$det(I)det(A)=det(B) \left[ det(I)=1\right]$$
$$det(A)=det(B)$$

### Relevant Questions Determine whether each of the following statements is true or false, and explain why.If A and B are square matrices of the same size, then AB = BA If $$A=\begin{bmatrix}-2 & 1&-4 \\-2 & 4&-1 \\ 1 &-1 &-4 \end{bmatrix} \text{ and } B=\begin{bmatrix}-2 & 4&2 \\-4 & -1&1 \\ 4 &1 &1 \end{bmatrix}$$
then AB=?
BA=?
True or false : AB=BA for any two square matrices A and B of the same size. If A and B are both $$n \times n$$ matrices (of the same size), then
det(A+B)=det(A)+det(B)
True or False? Let A and B be Hermitian matrices. Answer true or false for each of the statements that follow. In each case, explain or prove your answer. The eigenvalues of AB are all real. 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. Determine whether each statement is sometimes, always , or never true for matrices A and B and explain your reasoning. Stuck on one of them? Try a few examples.
a) If A+B exists , then A-B exists
b) If A and B have the same number of elements , then A+B exists. Let A,B and C be square matrices such that AB=AC , If $$A \neq 0$$ , then B=C.
Is this True or False?Explain the reasosing behind the answer. Mark each of the following statement true or false: If A and B are matrices such that AB = O and $$A \neq O$$, then B = O. Let A,B be two matrices with linearly independent columns . If $$Col(A)=Col(B)$$ then $$A(A^TA)^{-1}A^T=B(B^TB)^{-1}B^T$$ If A and B are $$n \times n$$ diagonalizable matrices , then A+B is also diagonalizable.