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15 maj 2024 · Every elementary matrix is invertible and its inverse is also an elementary matrix. In fact, the inverse of an elementary matrix is constructed by doing the reverse row operation on \(I\). \(E^{-1}\) will be obtained by performing the row operation which would carry \(E\) back to \(I\).
- Finding The Inverse of a Matrix
One way in which the inverse of a matrix is useful is to...
- Theorem
In order to do this, first recall some important properties...
- Finding The Inverse of a Matrix
17 wrz 2022 · First, we look at ways to tell whether or not a matrix is invertible, and second, we study properties of invertible matrices (that is, how they interact with other matrix operations). We start with collecting ways in which we know that a matrix is invertible.
To determine the inverse of an elementary matrix E, determine the elementary row operation needed to transform E back into I and apply this operation to I to nd the inverse.
In this section we introduce the concept of an elementary matrix. Elementary matrices are relatively simple objects, as their name suggests, but as we will see, they give us a simple method for understanding why our algorithm for computing the inverse of a matrix works.
Suppose \(B\) and \(C\) are two matrices that satisfy the properties of being an inverse of \(A\), i.e. \[ AB = BA = I \quad \text{and} \quad AC = CA = I. \] Then the following chain of identities proves that \(B\) and \(C\) must be equal:
What Are the Properties of Matrices for Inverse of a Matrix? The following are the important properties of the inverse of a matrix. The inverse of a matrix if it exists is unique. AB = BA = I. If matrix A is the inverse of matrix B, then matrix B is the inverse of matrix A.
It is clear that the corresponding matrices are inverses. Hence, every elementary matrix is invertible. Moreover, by using the socks and shoes property, we see that any product of invertible matrices is invertible, so that
