Studied by 14 people
The identity of a set/operation
The identity of a set/operation is the element of the set that when applied to any other element, leaves the other element unchanged
the identity under multiplication is 1
The identity under addition is 0
The identity under vector addition is the zero vector
The inverse of an element x
The inverse of the element x, in a set is the element that when applied to x, gives the identity
Example: the inverse of 5 under addition is -5
Ex: the inverse of 7 under multiplication is 1/7
Invertibility
Let A be an nxn matrix. A is invertible if there is an nxn matrix C such that:
CA = I
AC = I
where I is the nxn identity matrix
Singular matrix
A singular matrix is a matrix that is NOT invertible
Nonsingular matrix
A nonsingular matrix is an invertible matrix
Determining if Inverse (A^-1) of a 2×2 matrix exist
ad - bc is called the determinant of A
If A is an invertible nxn matrix
The equation Ax = b has the unique solution for every b in R^n
This can be done by applying A^-1 on both of the left sides of the equation
Ax = b
x = A^-1 b
Theorem about invertible matrices
If A is an invertible matrix, then A^-1 is invertible and
(A^-1) ^-1 = A
If A and B are nxn invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in reverse order
(AB)^-1 = B^-1 • A^-1
If A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of A^-1
(A^T)^-1 = (A^-1)^T
An elementary matrix
An elementary matrix is a matrix that is obtained by performing a single row operation on the identity matrix
An nxn matrix A is invertible if and only if…
A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A^-1
Algorithm for finding A^-1
To find the inverse of A using row-reduction:
