2.6 Matrices

Question 1

Matrix:
what are they?
when are they equal?

Answer 1

A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an
m × n matrix.
The plural of matrix is matrices.
A matrix with the same number of rows as
columns is called square.
Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal.

Question 2

Some terminology about matrices:

Answer 2

Let m and n be positive integers and let
A =
⎡ a11 a12 a1n ⎤
⎢ a21 a22 a2n ⎥
⎢ ⋅⋅ ⋅ ⎥
⎢ ⋅⋅ ⋅ ⎥
⎢ ⋅⋅ ⋅ ⎥
⎢ am1 am2 . amn ⎥
⎣ ⎦

The ith row of A is the 1 × n matrix [ai1, ai2, … , ain]. The jth column of A is the m × 1 matrix
⎡ a1j ⎤
⎢ a2j ⎢
⎢ . ⎢
⎢ ⎥
⎢ amj ⎥
⎣ ⎦
The (i, j)th element or entry of A is the element aij, that is, the number in the ith row and
jth column of A.
A convenient shorthand notation for expressing the matrix A is to write
A = [aij],
which indicates that A is the matrix with its (i, j)th element equal to aij.

Question 3

Matrix Addition:

Answer 3

Let A = [aij] and B = [bij] be m × n matrices. The sum of A and B, denoted by A + B, is
the m × n matrix that has aij + bij as its (i, j)th element. In other words, A + B = [aij + bij].

Matrices of different sizes cannot be added, because such matrices will not both have
entries in some of their positions.

Question 4

Matrix products:

Answer 4

A product of two matrices is defined only when the number of columns in the first matrix equals the number of rows of the second matrix.
Product not defined otherwise.

Let A be an m × k matrix and B be a k × n matrix. The product of A and B, denoted by AB, is
the m × n matrix with its (i, j)th entry equal to the sum of the products of the corresponding
elements from the ith row of A and the jth column of B. In other words, if AB = [cij], then
cij = ai1b1j + ai2b2j + ⋯ + aikbkj.

Question 5

Associativity of Matrix multiplication:

Answer 5

Matrix multiplication is associative, as can easily be proved using the associativity
of addition and multiplication of real numbers

Question 6

Matrix Commutativity.

Answer 6

Its not.

In general, suppose that A is an m × n matrix and B is an r × s matrix.
If n = r, AB defined. If s = m, BA defined.
Even when AB and BA are both defined,
they will not be the same size unless m = n = r = s. Hence, if both AB and BA
are defined and are the same size,
then both A and B must be square and of the same size.
Furthermore, even with A and B both n × n matrices, AB and BA are not necessarily equal.

Question 7

Identity Matrix:

Answer 7

The identity matrix of order n is the n × n matrix
In = [𝛿ij], (the Kronecker delta) where
𝛿ij = 1 if i = j and 𝛿ij = 0 if i ≠ j.
page 190, to see matrix.

Question 8

Matrix multiplied by identity Matrix:

Answer 8

Multiplying a matrix by an appropriately sized identity matrix does not change this matrix. In
other words, when A is an m × n matrix, we have
AIn = ImA = A.

Question 9

Power of Matrices:

Answer 9

Powers of square matrices can be defined because matrix multiplication is associative. When A
is an n × n matrix, we have
A^0 = In, A^r = AAA ⋯ A ( A written r times).

Question 10

Transpose of a matrix:

Answer 10

Let A = [aij] be an m × n matrix. The transpose of A, denoted by A^t,
is the n × m matrix
obtained by interchanging the rows and columns of A. In other words, if At = [bij], then
bij = aji for i = 1, 2, … , n and j = 1, 2, … , m.

ij th entry for A is ji the entry for its transpose.

Question 11

Symmetric Matrix:

Answer 11

A square matrix A is called symmetric if A = A^t.
Thus, A = [aij] is symmetric if aij = aji for
all i and j with 1 ≤ i ≤ n and 1 ≤ j ≤ n.

Note that a matrix is symmetric if and only if it is square and it is symmetric with respect to its
main diagonal (which consists of entries that are in the ith row and ith column for some i).

Question 12

Zero One Matrix.

Answer 12

A matrix all of whose entries are either 0 or 1 is called a zero–one matrix.

Question 13

Join of matrices A and B

Answer 13

Let A = [aij] and B = [bij] be m × n zero–one matrices. Then the join of A and B is the zero-one matrix with (i, j)th entry aij ∨ bij. The join of A and B is denoted by A ∨ B.

Join Or , Junior, Joiner.

Question 14

Meet of matrice A and B

Answer 14

Let A = [aij] and B = [bij] be m × n zero–one matrices.
The meet of
A and B is the zero–one matrix with (i, j)th entry aij ∧ bij. The meet of A and B is denoted
by A ∧ B.

Meet and Greet

Question 15

Boolean Product:

Answer 15

Let A = [aij] be an m × k zero–one matrix and B = [bij] be a k × n zero–one matrix. Then
the Boolean product of A and B, denoted by A ⊙ B, is the m × n matrix with (i, j)th entry cij
where

cij = (ai1 ∧ b1j) ∨ (ai2 ∧ b2j) ∨ ⋯ ∨ (aik ∧ bkj).

Question 16

Boolean powers of a square zero–one matrix:

Answer 16

Let A be a square zero–one matrix and let r be a positive integer. The rth Boolean power of
A is the Boolean product of r factors of A. The rth Boolean product of A is denoted by A^[r]
.
Hence,
A^[r] = A ⊙ A ⊙ A ⊙ ⋯ ⊙ A.

(This is well defined because the Boolean product of matrices is associative.) We also define
A^[0] to be In.

Question 17

Commutativity of matrix Addition?

Answer 17

A + B = B + A

Question 18

Matrix Addition associativity?

Answer 18

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

Question 19

Is matrix multiplication distributive over matrix addition?

Answer 19

(A + B)C = AC + BC

C(A + B) = CA + CB

Question 20

Matrix Multiplication’s associativity?

Answer 20

A(BC) = (AB)C.

Question 21

Diagonal Matrix:

Answer 21

The n × n matrix A = [aij] is called a diagonal matrix

if aij = 0 when i ≠ j.

Question 22

Transpose of

transpose of matrix is?

Answer 22

Matrix itself.

(A^t)^t = A.

Question 23

Matrix addition and multiplication involving transpose, 2 identities.

Answer 23

a) (A + B)^t = A^t + B^t

b) (AB)^t = B^tA^t

Question 24

The inverse of Matrix:

Answer 24

If A and B are n × n matrices with AB = BA = In, then B
is called the inverse of A (this terminology is appropriate because such a matrix B is unique) and A is said to be invertible.
The notation B = A^−1 denotes that B is the inverse of A.

Question 25

Inverse of
A
a b
c d

Answer 25

[I/(ad-bc) ]*

d -b
-c a

Question 26

Join and meet of same set?

Answer 26

A V A = A

A ^ A = A

Question 27

Commutativity of Meet and join?

Answer 27

A ∨ B = B ∨ A

B ∧ A = A ∧ B

Question 28

Associativity of meet and join operations:

Answer 28

(A ∨ B) ∨ C = A ∨ (B ∨ C).

A ∧ B) ∧ C = A ∧ (B ∧ C

Question 29

distributive laws of the meet over the

join operation

Answer 29

A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C).

A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C).