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1.1 Matrix and Properties

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Math 120 - Linear Algebra

Alexis Bernard A. Dalam, MS Math


CHAPTER 1: Part 1: Matrices
Formal Denition of Matrix
Denition 1.1. A matrix is a rectangular array of numbers called the entries, or elements, of the matrix.
Denition 1.2. The size of a matrix is a description of the numbers of rows and columns it has. A matrix is
called mn (pronounced m by n) if it has m rows and n columns.
A 1 m matrix is called a row matrix (a row vector), and n 1 matrix is called a column matrix (or
column vector).
Example 1.1. The following are examples of matrices.
1.

1 2
0 3

2.

2 2 3.2
7.3
2
3

3.

3
5
15

4.

2 3 5 2

5.

1

Remark 1.1. Matrices will usually be denoted by capital letters and the equation A = [a
ij
] means that the
element in the i-th row and j-th column of the matrix A equals a
ij
. The symbol A
mn
or [a
ij
]
mn
denotes the
matrix A with size mn. The general mn matrix has the form
A =

a
11
a
12
a
1n
a
21
a
22
a
2n
.
.
.
.
.
.
a
m1
a
m2
a
mn

Example 1.2. Make a 3 4 matrix A = [a


ij
] given the formula a
ij
=
2i
i + j
.
Denition 1.3. Given an mn matrix A = [a
ij
]
1. The diagonal entries of A are a
11
, a
22
, a
33
, . . .. It usually called the main diagonal if the matrix is an
n n matrix.
2. If m = n, then matrix A is called a square matrix i.e, an n n matrix.
3. A square matrix whose nondiagonal entries are all zero is called a diagonal matrix.
4. A diagonal matrix whose all of its diagonal entries are the same is called a scalar matrix.
5. If the scalar on the diagonal matrix is 1, the scalar matrix is called an identity matrix and denoted by I
n
if an n n matrix. That is, I
n
is an n n identity matrix of order n.
Example 1.3. Matrix A =

4 1
3 2

is a square matrix. Matrix B =

4 0 0
0 6 0
0 0 1

is a diagonal matrix. Matrix


C =

1 0 0
0 1 0
0 0 1

is an identity matrix.
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Denition 1.4. (Equality of Matrices) Matrices A, B are said to be equal if A and B have the same size
and their corresponding entries are equal. Thus, if A = [a
ij
]
mn
and B = [b
ij
]
rs
if and only if m = r and n = s,
and a
ij
= b
ij
, for all i and j.
Example 1.4. Consider the matrices A =

a b
c d

, B =

3 0
2 4

and C =

3 0 b
2 4 d

. Then A = B if and
only if a = 3, b = 0, c = 2 and d = 4. While A = C and B = C.
Example 1.5. Consider the matrices R =

2 4 6

and S =

2
4
6

. Despite the fact that R and S have the


same entries in the same order, R = S since R is 1 3 matrix and S is is 3 1 matrix.
Matrix Operations and Their Properties
Denition 1.5. (Addition of Matrices) Let A = [a
ij
] and B = [b
ij
] be of the same size (m n) matrices.
Then the sum A + B is an mn matrix obtained by adding corresponding entries of A and B; that is
A + B = [a
ij
] + [b
ij
] = [a
ij
+ b
ij
]
Example 1.6. Consider the matrices A =

2 4 0
3 6 5

, B =

4 1 2
4 0 2

and C =

4 2
3 2

. Find A + B,
A + C, and B + C.
Denition 1.6. (Scalar Multiplication of a Matrix) Let A = [a
ij
]is an mn matrix and c is a scalar, then
the scalar multiple cA is the mn matrix obtained by multiplying each entry of A by c; that is
cA = c[a
ij
] = [ca
ij
].
Example 1.7. Consider the matrices A =

2 4 0
3 6 5

and C =

4 2
3 2

. Find 2A and 3C.


Denition 1.7. (Additive Inverse of a Matrix) Let A = [a
ij
]is an mn matrix, then the additive inverse
A is the mn matrix obtained by replacing the elements of A by their additive inverses; that is
A = [a
ij
] = [a
ij
].
Denition 1.8. (Subtraction of Matrices) Let A = [a
ij
] and B = [b
ij
] be of the same size (mn) matrices.
Then the dierence of two matrices AB is an mn matrix obtained by subtracting corresponding entries of
B to A; that is
AB = [a
ij
] [b
ij
] = [a
ij
b
ij
]
Denition 1.9. (The Zero Matrix) A matrix all of whose entries are zero is called a zero matrix and denoted
by O (O
mn
if it is important to specify its size).
Theorem 1.1. (Algebraic Properties of Matrix Addition and Scalar Multiplication) Suppose A, B,
and C are mn matrix, and c and d are arbitrary scalars. Then
1. A + B = B + A (Commutativity)
2. (A + B) + C = A + (B + C) (Associativity)
2
3. A + O = A
4. A + (A) = O
5. c(A + B) = cA + cB, c(AB) = cAcB (Distributivity)
6. (c + d)A = cA + dA, (c d)A = cAdA (Distributivity)
7. c(dA) = (cd)A
8. 1A = A, OA = O, (1)A = A
9. cA = O = t = 0 or A = O
Denition 1.10. (Matrix Multiplication/ Matrix Product) If A = [a
ij
] is an mn matrix and B = [b
jk
]
is an n r matrix (the number of columns of A should be equal to the number of rows of B), then the product
AB is an mr matrix C = [c
ik
] whose (i, k)-th entries is dened by the formula
c
ik
=
n

j=1
a
ij
b
jk
= a
i1
b
1k
+ a
i2
b
2k
+ a
i3
b
3k
+ + a
in
b
nk
.
Example 1.8. Consider the matrices A =

1 2 1
2 1 1

and B =

4 0 3 1
5 2 1 1
1 2 0 6

. Find AB.
Theorem 1.2. (Algebraic Properties of Matrix Multiplication/Matrix Product) Suppose A, B, C and
D be matrices (whose sizes are such that the indicated operations can be performed) and let k be arbitrary scalar.
Then
1. A(BC) = (AB)C , (if A, B, C are, mn, n p, p q, respectively); (Associativity)
2. A(B + C) = AB + AC , (if B and C are mn, and A is p m); (Left Distributivity)
3. (B + C)D = BD + CD , (if B and C are mn, and D is n p); (Right Distributivity)
4. k(AB) = (kA)B = A(kB)
5. A(B) = (A)B = (AB)
6. I
m
A = A = AI
n
, if A is mn matrix (Multiplicative Identity)
Denition 1.11. (Matrix Powers) Let A = [a
ij
] be an n n matrix and k be any nonnegative integers, then
A
k
is dened as A
k
= AA A, of k factors. Thus, A
1
= A, and it is convenient to dene A
0
= I
n
.
Theorem 1.3. (Properties of Matrix Powers) If A is a square matrix and r and s are nonnegative integers,
then
1. A
r
A
s
= A
r+s
2. (A
r
)
s
= A
rs
Example 1.9. Consider the matrices A =

1 2
1 1

. Find A
2
and A
3
.
Example 1.10. Prove the following: If A and B are square matrices of the same size, then
(A + B)
2
= A
2
+ 2AB + B
2
3
Transpose of a Matrix and Its Properties
Denition 1.12. (Transpose of a Matrix) Let A = [a
ij
] be an mn matrix. The transpose of A, denoted
by A
T
, is the nm matrix obtained by interchanging the rows and columns of A. That is, the ith column of A
T
is the ith row of A, for all i
A
T
= [a
ji
]
Example 1.11. Consider the matrices A =

1 2 3
1 5 0

, B =

a b
c d

and C =

6 2 1

. Find A
T
,
B
T
and C
T
.
Theorem 1.4. (Properties of the Transpose) Let A and B be matrices ( whose sizes are such that the
indicated operations can be performed) and let k be a scalar. Then
1. (A
T
)
T
= A
2. (A + B)
T
= A
T
+ B
T
3. (kA)
T
= k(A
T
)
4. (AB)
T
= B
T
A
T
5. (A
r
)
T
= (A
T
)
r
, for all nonnegative integers r
Denition 1.13. A square matrix A is symmetric if A
T
= A. That is, if A is equal to its own transpose.
Example 1.12. Consider the matrices A =

1 3 2
3 5 0
2 0 7

and B =

2 4
2 3

. Determine whether the


following matrix are symmetric or not.
Theorem 1.5. (Properties of the Symmetric Matrix)
1. If A is a square matrix, then A + A
T
is a symmetric matrix.
2. For any matrix A, AA
T
and A
T
A are symmetric matrices.
Denition 1.14. A square matrix A is skew-symmetric if A
T
= A.
Example 1.13. Consider the matrices A =

0 3 1
3 0 2
1 2 0

and B =

1 2
2 3

. Determine whether the


following matrix are skew-symmetric or not.
Denition 1.15. A square matrix A is called upper triangular if all of the entries below the main diagonal
are zero. That is, an n n matrix A = [a
ij
] is an upper triangular if a
ij
= 0 if i > j.
Denition 1.16. The trace of an n n matrix A = [a
ij
] is the sum of the entries on its main diagonal and is
denoted by tr(A). That is,
tr(A) = a
11
+ a
22
+ + a
nn
.
Remark 1.2. (Partitioned Matrices) It will often be convenient to regard a matrix as being composed of a
number of smaller submatrices. By introducing vertical and horizontal lines into a matrix, we can partition it
into blocks..
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Assignment 1:
1. Consider the following matrices A =

1 0 2
1 3 2
4 2 3

, B =

6 8 5
4 2 7
3 1 2

, C =

1 3
2 4
5 2

,
D =

1 3
2 6

. Compute the indicated matrices (if possible).


(a) 3A2B
T
(b) (BC)
T
C
T
A
T
(c) (I
2
D)
3
(d) B(AC)
(e) C
T
C +
1
2
D
T
(f) (CD)C
T
2. Given that A =

1 2
3 4

and B =

1 0
1 1

, nd the value of X
(a) 2(A + 2B) = 3X
(b) 2(AB + X) = 3(X A)
3. Let A =

cos sin
sin cos

. Show that A
2
=

cos 2 sin 2
sin 2 2 cos

4. Find the 4 4 matrix A = [a


ij
] that satises the given condition a
ij
= sin

(i + j 1)
4

.
5. Find the 6 7 matrix A = [a
ij
] that satises the given condition a
ij
=

i + j, if i j
i j, if i > j
.
6. Find all 2 2 matrices A =

a b
c d

such that AB = BA, where B =

1 2
3 4

.
7. Prove: For a square matrices A and B, AB = BA if and only if (AB)(A + B) = A
2
B
2
.
8. Prove: If A and B are symmetric n n matrices, then A + B is symmetric.
9. Prove: If A is an n n matrix, then AA
T
is skew-symmetric.
10. Prove: If A and B are mn matrices, then (A + B)
T
= A
T
+ B
T
.
11. Prove: Show that there are no 2 2 matrices A and B such that AB BA = I
2
.
12. Prove: If A and B are n n matrices, then tr(A + B) = tr(A) + tr(B).
13. Consider the following matrix partition, A =

2 3 1 0
4 5 0 1

and B =

0 1 0
0 0 1
1 5 4
2 3 2

. Find AB, using


block multiplication.
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