There are different methods for determining the inverse of a matrix, including using an augmented matrix and row operations, using technology, and using a formula that includes calculating the reciprocal of the difference of the diagonals.
An augmented matrix
is formed from two given matrices by writing the columns of the first matrix to the left of a vertical line and the columns of the second matrix to the right of the vertical line. To determine the inverse of a square matrix
, start by augmenting
with the identity matrix of the same dimensions.
Next, perform a series of operations on the rows of the augmented matrix. Row operations
are operations that can be performed on a row of an augmented matrix to generate an equivalent matrix.
Matrix row operations:
1. Switch any two rows.
2. Multiply a row by a nonzero number.
3. Replace a row with the sum or difference of two rows.
The goal of performing row operations on the augmented matrix is to transform it into reduced row-echelon form, a form in which the left side is the identity matrix. When the augmented matrix is in this form, the right side will be the inverse of matrix .