An invertible matrix, also known as a non-singular matrix, is a square matrix that has a unique solution for its inverse. This means that it can be multiplied by another matrix to give the identity matrix. It is important because it allows for easier calculation and solves many practical problems in mathematics, engineering, and science.
A square matrix is invertible if its determinant is non-zero. The determinant is a numerical value that can be calculated by using the elements of the matrix. If the determinant is non-zero, then the matrix is invertible. If the determinant is zero, then the matrix is not invertible.
No, only square matrices (i.e. matrices with the same number of rows and columns) can be invertible. Non-square matrices do not have a unique solution for its inverse and therefore cannot be inverted.
The inverse of an invertible matrix can be found by using the Gauss-Jordan elimination method or by using the adjugate matrix method. Both methods involve transforming the matrix into a simpler form and then applying mathematical operations to find the inverse.
Invertible matrices are used in many areas of science and mathematics, such as solving systems of linear equations, calculating transformations in geometry, and solving differential equations. They also have practical applications in computer graphics, engineering, and economics.