What does it mean if two matrices have the same determinant?

A matrix’s determinant gives the volume of the parallelepiped whose sides are the columns of the matrix. If two matrices with identical dimensions have the same determinant, it means their corresponding parallelepipeds have the same volume.

Can 2 different matrices have the same determinant?

Thus, both the matrices have the same determinant value. Hence, we cay say, two different matrices can have the same determinant value.

How do you show that two matrices are similar?

Two matrices A and B are similar if there exists a nonsingular (invertible) matrix S such […] If 2 by 2 Matrices Satisfy A=AB−BA, then A2 is Zero Matrix Let A,B be complex 2×2 matrices satisfying the relation A=AB−BA.

What is the relation between trace and determinant?

The trace corresponds to the derivative of the determinant: it is the Lie algebra analog of the (Lie group) map of the determinant. This is made precise in Jacobi’s formula for the derivative of the determinant.

When A and B are similar matrices then?

Definition (Similar Matrices) Suppose A and B are two square matrices of size n . Then A and B are similar if there exists a nonsingular matrix of size n , S , such that A=S−1BS A = S − 1 B S .

When A and B are similar matrices then det A det B?

If A and B are similar, then A and B have the same determinant, rank and charac- teristic polynomial. Corollary 7. If A and B are similar, then they have the same eigenvalues, and A is invertible if and only if B is invertible. = det(P)det(B − λI) 1 det(P) = det(B − λI).

Do all matrices have the same determinants?

Every SQUARE matrix n×n has a determinant.

Does every matrix have a similar matrix?

For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically closed field), every matrix is similar to a matrix in Jordan form.

Are all diagonal matrices similar?

Although most matrices are not diagonal, many are diagonalizable, that is they are similar to a diagonal matrix. A matrix A is diagonalizable if A is similar to a diagonal matrix D. The following theorem tells us when a matrix is diagonalizable and if it is how to find its similar diagonal matrix D.

How can two similar matrices have the same rank?

Two similar matrices have the same rank, trace, determinant and eigenvalues. We start with a definition. Proposition A matrix is said to be similar to another matrix if and only if there exists an invertible matrix such that The transformation of into is called similarity transformation. The matrix is called change-of-basis matrix.

How do you prove that similar matrices have the same trace?

If A and B are n × n matrices of a field F, then show that trace ( A B) = trace ( B A). Hence show that similar matrices have the same trace. I’ve done the first part (proving that A B and B A have the same trace). I can show that here if you say so.

What is the importance of matrix similarity in linear algebra?

This type of matrices are very important for linear algebra. They are mainly used for diagonalizable matrices, since the method to diagonalize any matrix is based on the concept of matrix similarity. The diagonalization of a matrix consists of calculating a similar matrix that, at the same time, is a diagonal matrix.

What is the characteristic polynomial of two similar matrices?

The characteristic polynomial and the minimum polynomial of two similar matrices are the same. A matrix and its transpose are similar. If matrices A and B are similar, matrix B can be found by applying elementary operations on the rows of matrix A, and vice versa. Obviously, the similarity between matrices is a reflective operation.