1.1 Matrix and Properties
1.1 Matrix and Properties
1.1 Matrix and Properties
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
4 1
3 2
4 0 0
0 6 0
0 0 1
1 0 0
0 1 0
0 0 1
is an identity matrix.
1
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
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
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
0 3 1
3 0 2
1 2 0
and B =
1 2
2 3
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
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
(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
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