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University of Waterloo

Mathematics

MATH 136

Robert Andre

Spring

Description

Friday, January 31 − Lecture 12 : Matrices and matrix algebra
Concepts:
1. matrix and the operation of addition, multiplication, scalar multiplication and
transposition on matrices.
2. rules of matrix algebra.
3. zero matrix
4. matrix-vector multiplication.
5. system of linear equations expressed as a matrix equation.
12.1 Definition − Matrices. If m and n are positive integers, a matrix of size m × n (or of
dimension m × n), is a rectangular array of real numbers, arranged inm rows and n
column. It is represented as
A = [a ]ij m × n
The a ’ijare called the entries orcomponents of the matrix. The symbol i is called the
row index or the entrée and j is called its column index.
Example : This is a 3 by 3 matrix A :
12.1.1 Two matrices are said to be equalmatrices if and only if their corresponding
entries are equal. Because of this matrices of different dimensions cannot be equal.
12.1.2 Remark − A vector, such as u = (2, 1, 7) for example, can be also be
expressed in matrix form, u = [2, 1, 7] 1 × 3, or
The form it should take is usually determined by the context.
12.2 Definition − If A = [a ] ij m×n then the vectors r = ia , ai1 .i2, a ) iinℝ are called the
n
row-vectors of A (1 ≤ i ≤ m) . The vectors c j= (a 1ja 2j..., anj in ℝ are called the
column-vectors of A (1 ≤ j ≤ n). So, when convenient, we can think of A as a pile of row-
vectors, r , r , ..., r all stacked one on top the other, or as a juxtaposition of column
1 2 m
vectors c ,1c 2 ..., cn: 12.3 Definitions − A matrix that contains only zeroes as entries is called the zero matrix.
12.4 Operations with matrices − Let M m×n denote the set of all m× n matrices with real
numbers as entries. Then will define addition, scalar multiplicationand transposition on
matrices in M m×n as follows:
Operation What we start with : What we get :
Addition
Scalar
multiplication
Transposition
Suppose
then from the definition we see that the transpose of A exchange rows with columns: 12.4.1 Example – Let Aand B be the given matrices:
T
a) Compute 3A – 2B
Answer:
b) Find the (2, 3) entry of C = A + 3B.
T
Answer: Let C = [c ], ij= [b ], ij= A = [d ]. ij
c23 d + 23 = a 233b =327 + 3(23 = −7
th T
c) Find the (1, 3) entry of H = (2A + B)
Answer: 4
th T
d) Find the (2, 2) entry of E = 2A + A
12.4.2 Theorem – Let m and n be fixed positive integers. Then the set M m × nof all
m × n matrices with real number entries equipped with + and scalar multiplication
described above satisfiesthe following basic properties:
1) M is closed under addition and scalar multiplication.
m × n
2) Addition in M is associative and commutative.
m × n
3) Matrices in M distribute over finite sums of scalars.
m × n
4) Scalars distribute over finite sums of matrices.
5) Transpose properties:
i) (A ) = A,
ii) (A + B) = A + B , T
iii) (cA)T = cA T
Proof : The proof of these follow immediately from the definition of + and scalar
multiplication and transposition and so is omitted. 12.5 Some special matricesand matrix related definitions
T
The square matrix A is said to be symmetric if A = A . That is, if A = [a ] ij n×nhen a ija .ji
A column matrix is a matrix which has only one column. While a row matrix is a
matrix with only one row. Both a column matrix and row matrix can be referred to as
vectors when useful and appropriate.
Given a square matrix A , tn×ntrace of the matrix

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