## 16.4 Outline

- Matrix operations
- matrix equation
- equality
- addition
- multiplication by a scalar
- subtraction
- multiplication
- noncomformable

- Zero-one matrices
- Algebraic properties of matrices
- zero matrix
- identity matrix for multiplication
- main diagonal
- additive inverse
- multiplicative inverse
- properties
- commutative
- associative
- identity
- inverse
- distributive

- Inverse property
- nonsingular
- procedure for finding
- Systems of equations
- inverse method for finding solution
- calculator usage

## 16.4 Essential Ideas

**Matrix Operations**

*Equality* [M] = [N] if and only if matrices [M] and [N] are the same order and the corresponding entries are the same.

*Addition* [M] + [N] = [S] if and only if [M] and [N] are the same order and the entries of [S] are found by adding the corresponding entries of [M] and [N].

*Multiplication by a scalar* *c*[M] = [M] is the matrix in which each entry of [M] is multiplied by the scalar (real number) *c*.

*Subtraction* [M][N] = [D] if and only if [M] and [N] are the same order and the entries of [D] are found by subtracting the entries of [N] from the corresponding entries of [M].

*Multiplication* Let [M] be an *m* x *r* matrix and [N] an *r* x *n *matrix. The product matrix [M][N] = [P] is an *m* x *n* matrix. The entry in the *i *th row and *j *th column of [M][N] is the sum of the products formed by multiplying each entry of the *i *th row of [M] by the corresponding entry in the *j *th column of [N].

**Properties of Matrices ??? **

Property | Addition | Multiplication |

Commutative | [M] + [N] = [N] + [M] | [M][N] does not equal [N][M] |

Associative | ([M] + [N]) + [P] = [M] + ([N] + [P]) | ([M][N])[P] = [M]([N][P]) |

Identity | [M] + [0] = [0] + [M] | [I][M] = [M][I] = [M] |

Inverse | [M] + [M] = [M] + [M] =[0] | [M][M] = [M][M] =[I] |

Distributive | [M]([N] + [P]) = [M][N] + [M][P] | |

Distributive | ([N] + [P])[M] = [N][M] + [P][M] |

**Inverse of a Matrix **If [A] is a square matrix and if there exists a matrix [A]

^{-1}such that [A]

^{-1}[A] = [A][A]

^{-1}= [I] where [I] is the identity matrix for multiplication, then [A]

^{-1}is called the inverse of [A] for multiplication.

**Procedure for Finding the Inverse of a Matrix**

To find the inverse of a *square* matrix *A*:

**Step 1 :** Augment [A] with [I]; that is write [A | I], where [I] is the identity matrix of the same order as [A].

**Step 2: ** Perform elementary row operations using Gauss-Jordan elimination to change the matrix [A] into the identity matrix [I], if possible.

**Step 3: ** If at any time you obtain all zeros in a row or column to the left of the dividing line, then there will be no inverse.

**Step 4: ** If steps 1 and 2 can be performed, the result in the augmented part is the inverse of [A].