pöördmatriisi

Pöördmatriisi, also known as the inverse matrix, is a fundamental concept in linear algebra. For a square matrix A, its inverse, denoted as A⁻¹, is another matrix such that when multiplied by A, the result is the identity matrix I. Mathematically, this is expressed as A * A⁻¹ = A⁻¹ * A = I. The identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere.

Not all square matrices have an inverse. A matrix that has an inverse is called invertible or non-singular. A matrix that does not have an inverse is called singular. A key determinant for invertibility is the determinant of the matrix. A square matrix is invertible if and only if its determinant is non-zero.

The concept of the inverse matrix is crucial for solving systems of linear equations. If we have a system of equations represented in matrix form as Ax = b, where A is the coefficient matrix, x is the vector of variables, and b is the constant vector, then if A is invertible, we can solve for x by multiplying both sides by A⁻¹: x = A⁻¹b.

The calculation of the inverse matrix can be done through various methods, including Gaussian elimination (also known as Gauss-Jordan elimination) or using the adjugate matrix. The adjugate method involves calculating the determinant and the matrix of cofactors. Inverse matrices have applications in cryptography, computer graphics, and many scientific and engineering fields.