OhMyMarkov said:Hello everyone!
I'm struggling to find a general formula for obtaining an inverse of a symmetric matrix, for e.g.
1 i -1
i -i 2
-1 2 1
Any help is appreciated!
The inverse of a symmetric matrix is a matrix that, when multiplied by the original symmetric matrix, results in an identity matrix. In other words, it "undoes" the original matrix, just like how division is the inverse operation of multiplication.
The inverse of a symmetric matrix can be calculated by using a mathematical formula known as the "Gauss-Jordan elimination method". This involves transforming the original matrix into reduced row echelon form and then back into matrix form.
Some important properties of an inverse of a symmetric matrix include: it is unique, it is also symmetric, and the inverse of the inverse is the original matrix. Additionally, if the original matrix is invertible (i.e. its determinant is not equal to 0), then its inverse is also invertible.
The inverse of a symmetric matrix is important in various areas of mathematics, engineering, and science. It is used to solve systems of linear equations, calculate determinants, and find solutions to optimization problems. It also plays a crucial role in the study of eigenvectors and eigenvalues.
No, not all matrices have an inverse. In order for a matrix to have an inverse, it must be square (i.e. have the same number of rows and columns) and its determinant must not equal 0. A matrix that does not have an inverse is called a singular or non-invertible matrix.