Matrices and Determinants

Question 1

Who gave the idea of matrices? When?

Answer 1

Arthur Cayley, an English mathematician of the nineteenth century, first developed the “Theory of Matrices” in 1858.
James Sylvester

Question 2

what is a matrix?

Answer 2

a matrix (plural: matrices) is a rectangular array or formation of a collection of real numbers, arranged in rows and columns and enclosed by brackets “[ ]”. The real numbers used in the formation of a matrix are termed as entries or elements of a matrix.
The matrices are denoted by the capital letters of the English alphabets.

Question 3

what is the order of the matrix?

Answer 3

The number of rows and columns in a matrix specifies its order. If a matrix G has m rows and n columns then the order of the matrix is m-by-n.
In boards, explain with an example.

Question 4

what are equal matrices?
Explain with two examples.

Answer 4

Matrix A & B are two equal matrices if only
1) the order of A=the order of B
2) their corresponding entries are equal.
pg 3

Question 5

Row matrix?

Answer 5

a matrix that has only one row.
example: with a board pattern

Question 6

column matrix?

Answer 6

a matrix that has only one column
example: with a board pattern

Question 7

rectangular matrix?

Answer 7

A matrix M is called rectangular if, the number of rows of M is not equal to the no. of columns of N
example: with a board pattern

Question 8

square matrix?

Answer 8

no. of rows = no. of columns
example: with a board pattern

Question 9

Null or zero matrices?

Answer 9

each of its entries is 0
example: with a board pattern

Question 10

A null matrix is represented by _____.

Answer 10

O

Question 11

null matrix is also called

Answer 11

zero matrix

Question 12

what is the transpose of a matrix?

Answer 12

A matrix obtained by changing the rows into columns and columns into rows of a matrix is called the transpose of that matrix.
Transpose of a matrix (let’s suppose matrix A) is denoted by At.
example: with a board pattern in the book.

Question 13

negative of a matrix?

Answer 13

the negative of a matrix is obtained by changing the signs of all the entries. If B is a matrix, then its negative is denoted by -B
example: with a board pattern in book

Question 14

Symmetric Matrix

Answer 14

A square matrix is symmetric if it is equal to its transpose. Example matrix A is symmetric if A=At
example: with a board pattern in the book

Question 15

Skew symmetric matrix

Answer 15

A square matrix is skew-symmetric if the transpose of a matrix is equal to its negative.
example: with a board pattern in the book

Question 16

Diagnol matrix?

Answer 16

A square matrix is called a diagonal matrix if at least one of the entries of its diagonal is not zero and non-diagonal entries are zero.
example: with a board pattern in the book

Question 17

scalar matrix?

Answer 17

a diagonal matrix is called a scalar matrix, if all the diagonal entries are the same.
example: with a board pattern in the book
shouldn’t be 1 or 0

Question 18

identity matrix?

Answer 18

an identity matrix is a diagonal matrix if all diagonal entries are 1
example: with a board pattern in the book

Question 19

what is a unit matrix represented by?

Answer 19

I

Question 20

pattern for the order of a matrix

Answer 20

R by C

Question 21

what is the condition for addition & subtraction of matrices?

Answer 21

same order

Question 22

what won’t change after adding or subtracting matrices?

Answer 22

order of matrices will remain the same to its original matrix

Question 23

what is the commutative law of the addition of matrices?

Answer 23

A+B=B+A

Question 24

what is the associative law of the addition of matrices?

Answer 24

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

Question 25

what is the additive identity of a matrix?

Answer 25

For any matrix A and zero matrix O of the same order, O is called the additive identity of a matrix A as A+O=A=O+A
2+ ?= 2
2+0=2
?=0

Question 26

what is the additive inverse of a matrix?

Answer 26

The additive inverse of any matrix is obtained by changing to negative, the symbols of each non-zero entry of A.
A+B=O=B+A
2+ ?=0
2+ (-2)=0
?= -2

Question 27

what is the associative law of multiplication?

Answer 27

(AB)C=A(BC)

Question 28

what is the distributive law of multiplication over addition?

Answer 28

1) A(B+C)=AB+AC (left dist. law)
2) (A+B)C= AC+BC (right dist. law)

Question 29

what is the distributive law of multiplication over subtraction?

Answer 29

1) A(B-C)=AB-AC
2) (A-B)C=AC-BC

Question 30

what is the commutative law of multiplication of matrices?

Answer 30

commutative law of multiplication is not always true
AB=BA
AB is not equal to BA
pg 18

Question 31

what is the multiplicative identity of a matrix?

Answer 31

AB=A=BA
where B is an identity matrix (diagonally 1)
2 x ?= 2
2x1=2
?=1

Question 32

Adj of a matrix using a long method?

Answer 32

first, find cofactor
then transpose

Question 33

what is a singular matrix?

Answer 33

A square matrix A is called singular matrix if the|A| is = 0

Question 34

what is a non-singular matrix?

Answer 34

A square matrix A is called non-singular matrix if the|A| is ≠ 0

Question 35

Adj of a matrix using a short method?

Answer 35

interchanging the diagonal entries and changing the sign of the entries.

Question 36

what is the multiplicative inverse of a matrix?

Answer 36

the inverse of A is denoted by A-1
2 x ? = 1
2 x 1/2 = 1 (identity matrix)
?=1/2
first, find A-1 and multiply it with A to get the identity matrix.

Question 37

what is the formula for the inverse of k -1?

Answer 37

1 x Adj k / det k

Question 38

what is det of matrix?

Answer 38

ad-bc

Question 39

how can a simultaneous linear equation be found using matrices?

Answer 39

2 methods
Matrix inversion method
Cramer’s rule