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1

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

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2

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

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3

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

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4

Singular matrix

A singular matrix is a matrix that is *NOT* invertible

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5

Nonsingular matrix

A nonsingular matrix is an invertible matrix

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6

Determining if Inverse (A^-1) of a 2×2 matrix exist

ad - bc is called the *determinant *of **A**

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7

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**

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8

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

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9

An elementary matrix

An elementary matrix is a matrix that is obtained by performing a single row operation on the identity matrix

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10

An nxn matrix **A **is invertible if and only if…

**A **is row equivalent to **I**n, and in this case, any sequence of elementary row operations that reduces **A **to **I**n also transforms **I**n into **A**^-1

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11

Algorithm for finding **A**^-1

To find the inverse of **A** using row-reduction:

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