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# Algebraic Properties of Matrix Operation

**Recall the Properties of Real Numbers:
• Match the following**

Commutative for multiplication | |

Associative for multiplication | 3 (5 + 4) = 3(5) + 3(4) |

Multiplicative identity | |

Distributive | 5+(–5)=0 |

Identity for multiplication | 5 + 4 = 4 + 5 |

Commutative for addition | |

Associative for addition | 5 ( 1/5) = 1 |

Identity for addition | 5 + 0 = 5 |

Additive identity | (5 + 4) + 3 =5 + (4 + 3) |

**• Of the above, which do you think hold true for matrices?
**Commutative for multiplication

Associative for multiplication

Multiplicative identity

Distributive

Identity for multiplication

Commutative for addition

Associative for addition

Identity for addition

Additive identity

**Properties of Matrix Operations:
**•

**Theorem 7:**The following are true for matrices (recall that size of the matrices are important):

A + B = B + A

(A + B) + C = A + (B + C)

There exists a unique matrix O (called the zero matrix) such that A + O = A

There exists a unique matrix P such that A + P = O

• We have already seen that AB = BA is NOT true for matrices

**• Theorem 8: **The following are also true for matrices (recall that size of the
matrices are important)

A(BC) = (AB)C

For scalars r and s, r(sA) = (rs)A = s(rA)

r(AB) = (rA)B = A(rB)

**• Theorem 9: **The following are also true for matrices (recall that size of the
matrices are important)

(A + B)C = AC + BC

A(B + C) = AB + AC

(r + s)A = rA + sA

r(A + B) = rA + rB

• Page 69, Exercises 2 and 4

**Transpose of a Matrix:
**• For any element a

_{ij}in a matrix A, the transpose, denoted A

^{T}= (b

_{ij}), where b

_{ij}= a

_{ji}for all i,j

• For example, for the matrix ,

**• Theorem 10:**

(A + B)^{T} = A^{T} + B^{T}

(AC^)T = C^{T} A^{T}

(A^{T})^{T} = A

• A matrix A is symmetric if A = A^{T}

• An nxn matrix is called a square matrix

• In a matrix, the values a_{ii} are called the main diagonal

• Page 69, Exercise 8

**The Identity Matrix:
**• The nxn identity matrix, denoted as I

_{n}is the square matrix with 1’s on the main diagonal and 0’s everywhere else.

• For example,

• The identity matrix is the multiplicative identity for matrix multiplication.
This means that

A I_{n} = A = I_{n} A

**Scalar Products and Vector Norms:
**• A vector is an nx1 matrix, usually denoted with
and
.

NOTE: the book uses bold notation for vectors.

• The scalar product for two vectors is simply matrix multiplication

• In a similar fashion,

• The norm is given by

• For two vectors, the Euclidean distance is given by

• Page 69, Exercises 14 and 20