Matrices and Determinants Flashcards
Who gave the idea of matrices? When?
Arthur Cayley, an English mathematician of the nineteenth century, first developed the “Theory of Matrices” in 1858.
James Sylvester
what is a matrix?
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.
what is the order of the matrix?
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.
what are equal matrices?
Explain with two examples.
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
Row matrix?
a matrix that has only one row.
example: with a board pattern
column matrix?
a matrix that has only one column
example: with a board pattern
rectangular matrix?
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
square matrix?
no. of rows = no. of columns
example: with a board pattern
Null or zero matrices?
each of its entries is 0
example: with a board pattern
A null matrix is represented by _____.
O
null matrix is also called
zero matrix
what is the transpose of a matrix?
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.
negative of a matrix?
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
Symmetric Matrix
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
Skew symmetric matrix
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
Diagnol matrix?
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
scalar matrix?
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
identity matrix?
an identity matrix is a diagonal matrix if all diagonal entries are 1
example: with a board pattern in the book
what is a unit matrix represented by?
I
pattern for the order of a matrix
R by C
what is the condition for addition & subtraction of matrices?
same order
what won’t change after adding or subtracting matrices?
order of matrices will remain the same to its original matrix
what is the commutative law of the addition of matrices?
A+B=B+A
what is the associative law of the addition of matrices?
(A+B)+C=A+(B+C)
what is the additive identity of a matrix?
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
what is the additive inverse of a matrix?
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
what is the associative law of multiplication?
(AB)C=A(BC)
what is the distributive law of multiplication over addition?
1) A(B+C)=AB+AC (left dist. law)
2) (A+B)C= AC+BC (right dist. law)
what is the distributive law of multiplication over subtraction?
1) A(B-C)=AB-AC
2) (A-B)C=AC-BC
what is the commutative law of multiplication of matrices?
commutative law of multiplication is not always true
AB=BA
AB is not equal to BA
pg 18
what is the multiplicative identity of a matrix?
AB=A=BA
where B is an identity matrix (diagonally 1)
2 x ?= 2
2x1=2
?=1
Adj of a matrix using a long method?
first, find cofactor
then transpose
what is a singular matrix?
A square matrix A is called singular matrix if the|A| is = 0
what is a non-singular matrix?
A square matrix A is called non-singular matrix if the|A| is ≠ 0
Adj of a matrix using a short method?
interchanging the diagonal entries and changing the sign of the entries.
what is the multiplicative inverse of a matrix?
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.
what is the formula for the inverse of k -1?
1 x Adj k / det k
what is det of matrix?
ad-bc
how can a simultaneous linear equation be found using matrices?
2 methods
Matrix inversion method
Cramer’s rule
