Chapter 1 Definitions

Question 1

What is a matrix?

Answer 1

A matrix is a rectangular array of objects

Question 2

What is a reduced matrix?

Answer 2

A matrix is said to be reduced if each of the following conditions holds:

(i) the first nonzero entry in each row is 1, and in the column where this leading 1 occurs, all other entries are zero.
(ii) the leading one in each row lies to the right of the leading one in the preceding row
(iii) all rows consisting entirely of zeros lies at the bottom of the matrix

Question 3

What are equal matrices?

Answer 3

Two matrices are equal if and only if they contain precisely the same entries

Question 4

Row equivalent?

Answer 4

When one of the elementary row operations is used, we say the resulting matrix is row equivalent. Denoted by ~

Question 5

What is a homogeneous system?

Answer 5

All constants(c) are 0

Question 6

Theorem 1

Answer 6

Every homogeneous system is consistent

Question 7

Theorem 2

Answer 7

A homogeneous system has an infinite number of solutions if n>m (unknowns>equations)

Question 8

What is a square matrix?

Answer 8

where the number of rows equals the number of columns

Question 9

What is a main diagonal?

Answer 9

(read L-R) the diagonal of a square matrix consists of the following entries (a[11], a[22],…,a[nm].

Question 10

What is an Identity Matrix?

Answer 10

An identity matrix of order n is a matrix with 1’s on the main diagonal and zeros elsewhere, denoted by I[n]

Question 11

Communative

Answer 11

A and B are called communative (AB=BA) if and only if AB=BA.

Question 12

Zero Matrix

Answer 12

a matrix consisting entirely of zeros

Question 13

Involutory

Answer 13

A is called involutory if and only if A^2 = I

Question 14

Idempotent

Answer 14

A is called idempotent if and only if A^2 = A

Question 15

Nilpotent

Answer 15

A called nilpotent if A^k = 0 matrix, for some positive integer k, the smallest such value of k is called index of nilpotency.

Question 16

Diagonal Matrix

Answer 16

An n-square matrix is called diagonal if the entries above and below the diagonal are zeros.

Question 17

Upper Triangular

Answer 17

An n-square matrix is called upper triangular if the entries below the main diagonal are zeros.

Question 18

Lower Triangular

Answer 18

a lower triangular matrix consists of all zero entries above the main diagonal

Question 19

Triangular

Answer 19

the matrix is upper and lower triangular

Question 20

Theorem 3

Answer 20

i. A+0=A
II.0A=0
III.Commutative Law for matrix addition
IV.Associative law for matrix addition
V.Distributive law for scalar multiplication
VI. 1A = A
VII.(a+b)A=aA+bA

Question 21

Commutative Law for matrix addition

Answer 21

A + B = B + A

Question 22

Associative Law for matrix addition

Answer 22

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

Question 23

Distributive Law for Scalar Multiplication

Answer 23

a(A+B)=aA+aB

Question 24

Nonsingular

Answer 24

If the inverse of A exists, A is called invertible, or nonsingular. A^-1

Question 25

Theorem 4

Answer 25

Suppose A and B are nxn invertible matrices.

Then AB is invertible and (AB)^-1 = B^-1A^-1

Question 26

Theorem 5

Answer 26

Suppose a sequence of matrices are all nxn invertible matrices. The inverse of the sequence is the inverses of all the matrices in reverse order

Question 27

Theorem 6

Answer 27

Suppose A is invertible. Then A^-1 is also invertible, and (A^-1)^-1=A

Question 28

Trace

Answer 28

Let A be an n-square matrix. The trace of A is the sum of the diagonal entries, denoted by tr(A).

Question 29

Transpose

Answer 29

Let A be an mxn matrix. The transpose of A is the matrix obtained by making each row of A a column, while preserving the order, denoted by A^t

Question 30

Theorem 7

Answer 30

Let A and B be mxn matrices.

(i) (A^t)^t = A
(ii) (A+B)^t = A^t + B^t
(iii) (AB)^t = B^tA^t
(iv) If A is invertible, then (A^-1)^t = (A^t)^-1

Question 31

Symmetric

Answer 31

Let A be n-square matrix. A is called symmetric if A^t=A

Question 32

Skew-Symmetric

Answer 32

A is called skew-symmetric if A^t=-A

Question 33

Theorem 8

Answer 33

Let A and B be n-square symmetric matrices. AB is symmetric if and only if A and B commute

Question 34

Elementary Matrix

Answer 34

An n-square matrix is called an elementary matrix if it can be obtained from the Identity matrix by means of a single elementary row operation.

Question 35

Theorem 9

Answer 35

Every elementary matrix is invertible

Question 36

Theorem 10

Answer 36

An n-square matrix is invertible if and only if it is the product of elementary matrices.