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MATRICES
Definition of a Matrix
A rectangular array of numbers is called a matrix. The horizontal arrays of a matrix are called its rows
and the vertical arrays are called its columns. A matrix having m rows and n columns is said to have
the order m × n. A matrix A of order m × n can be represented in the following form:
ଵଵ
ଵଶ
ଵଷ
ଵ
ଶଵ
ଶଶ
ଶଷ…
ଶ
ଷଵ
ଷଶ
ଷଷ…
ଷ
ଵ
ଶ
ଷ…
Matrix is enclosed by [ ] or ( ). Compact form the above matrix is represented by [a ij
]
m x n
or A = [a ij
].
i. Element of a Matrix The numbers a 11
, a 12
… etc., in the above matrix are known as the
elements of the matrix, generally represented as a ij
, which denotes element in i th
row and j th
column.
ii. Order of a Matrix In above matrix has m rows and n columns, then A is of order m x n.
Types of Matrices
- Row Matrix A matrix having only one row and any number of columns is called a row matrix.
- Column Matrix A matrix having only one column and any number of rows is called column
matrix.
- Rectangular Matrix A matrix of order m x n is called rectangular matrix.
- Horizontal Matrix A matrix in which the number of rows is less than the number of columns,
is called a horizontal matrix.
- Vertical Matrix A matrix in which the number of rows is greater than the number of columns,
is called a vertical matrix.
- Null/Zero Matrix A matrix of any order, having all its elements are zero, is called a null/zero
matrix. i.e., a ij
= 0, ∀ i, j.
- Square Matrix A matrix of order m x n, such that m = n, is called square matrix i.e, number of
rows equal to number of columns.
൩ Principal/leading Diagonal
NOTE:- In a square matrix the diagonal from left hand side upper corner to right hand side
lower corner is known as leading diagonal or principal diagonal.
- Diagonal Matrix A square matrix A = [a ij
]
m x n
, is called a diagonal matrix, if all the elements
except those in the leading diagonals are zero, i.e., a ij
= 0 for i ≠ j. It can be represented as A =
diag[a 11
a 22
… a nn
]
- Scalar Matrix A square matrix in which every non-diagonal element is zero and all diagonal
elements are equal, is called scalar matrix. i.e., in scalar matrix a ij
= 0, for i ≠ j and a ij
= k, for i =
j.
- Unit/Identity Matrix A square matrix, in which every non-diagonal element is zero and every
diagonal element is 1, is called, unit matrix or an identity matrix.
- Upper Triangular Matrix A square matrix A = [a ij
]
n x n
is called a upper triangular matrix, if [a ij
]
= 0, ∀ i > j.
- Lower Triangular Matrix A square matrix A = [a ij
]
n x n
is called a lower triangular matrix, if [a ij
]=
0, ∀ i < j.
- Sub matrices A matrix which is obtained from a given matrix by deleting any number of rows
or columns or both is called a sub matrix of the given matrix.
- Equal Matrices Two matrices A and B are said to be equal, if both having same order and
corresponding elements of the matrices are equal.
- Singular Matrix A square matrix A is said to be singular matrix, if determinant of A denoted by
det (A) or |A| is zero, i.e., |A|= 0, otherwise it is a non-singular matrix.
Algebra of Matrices
- Addition of Matrices
Let A and B be two matrices each of order m x n. Then, the sum of matrices A + B is defined only if
matrices A and B are of same order. If A = [a ij
]
m x n
, A = [a ij
]
m x n
Then, A + B = [a ij
]
m x n
Properties of Addition of Matrices If A, B and C are three matrices of order m x n, then
i. Commutative Law A + B = B + A
ii. Associative Law (A + B) + C = A + (B + C)
iii. Existence of Additive Identity A zero matrix (0) of order m x n (same as of A), is additive
identity, if A + 0 = A = 0 + A
iv. Existence of Additive Inverse If A is a square matrix, then the matrix (- A) is called additive
inverse, if A + ( – A) = 0 = (- A) + A
v. Cancellation Law A + B = A + C ⇒ B = C (left cancellation law),
B + A = C + A ⇒ B = C (right cancellation law)
- Subtraction of Matrices
Let A and B be two matrices of the same order, then subtraction of matrices, A – B, is defined as A – B =
[a ij
]
m x n,
where A = [a ij
]
m x n
, B = [b ij
]
m x n
- Multiplication of a Matrix by a Scalar
Let A = [a ij
]
m x n
be a matrix and k be any scalar. Then, the matrix obtained by multiplying each element
of A by k is called the scalar multiple of A by k and is denoted by kA, given as kA= [ka ij
]
m x n
Properties of Scalar Multiplication If A and B are matrices of order m x n, then
i. k(A + B) = kA + kB
ii. (k 1
)A = k 1
A + k 2
A
iii. k 1
k 2
A = k 1
(k 2
A) = k 2
(k 1
A)
iv. (- k)A = – (kA) = k( – A)
- Multiplication of Matrices
Let A = [a ij
]
m x n
and B = [b ij
]
n x p
are two matrices such that the number of columns of A is equal to the
number of rows of B, then multiplication of A and B is denoted by [AB] m x p
Properties of Multiplication of Matrices
i. Commutative Law Generally AB ≠ BA
ii. Associative Law (AB)C = A(BC)
iii. Existence of multiplicative Identity A.I = A = I.A, where I is called multiplicative Identity.
iv. Distributive Law A(B + C) = AB + AC
v. Cancellation Law If A is non-singular matrix, then
AB = AC ⇒ B = C (left cancellation law)
BA = CA ⇒B = C (right cancellation law)
vi. AB = 0, does not necessarily imply that A = 0 or B = 0 or both A and B = 0.
Important Points to be remembered
i. If A and B are square matrices of the same order, say n, then both the product AB and BA are
defined and each is a square matrix of order n.
ii. In the matrix product AB, the matrix A is called premultiplier (prefactor) and B is called
postmultiplier (postfactor).
iii. The rule of multiplication of matrices is row column wise (or → ↓ wise) the first row of AB is
obtained by multiplying the first row of A with first, second, third,… columns of B respectively;
similarly second row of A with first, second, third, … columns of B, respectively and so on.
Trace of a Matrix
The sum of the diagonal elements of a square matrix A is called the trace of A, denoted by
trace(A) or tr(A).
Properties of Trace of a Matrix
- Trace (A ± B)= Trace (A) ± Trace (B)
- Trace (kA)= k Trace (A)
- Trace (A’ ) = Trace (A)
- Trace (O) = 0
- Trace (AB) ≠ Trace (A) x Trace (B)
- Trace (AA’) ≥ 0
DETERMINANT
The determinant is a number associated with any square matrix;
We’ll write it as 𝑑𝑒𝑡 𝐴 𝑜𝑟 |𝐴|.
The determinant encodes a lot of information about the matrix;
The matrix is invertible exactly when the determinant is non-zero.
Determinant of a 2 × 2 matrix
The determinant of the matrix 𝐴 = ቂ
ቃ is denoted by ቚ
ቚ (note the change from square
brackets to vertical lines) and is defined to be the number 𝑎𝑑 − 𝑏𝑐.
That is: ቚ
Properties
- If you exchange two rows of a matrix, you reverse the sign of its determinant from positive to
negative or from negative to positive.
Property 1 tells us that ቚ
ቚ = 1 & Property 2 tells us that ቚ
- (a) If you multiply one row of a matrix by k, the determinant is multiplied by k:
(b) The determinant behaves like a linear function on the rows (columns) of the matrix:
From these three properties (1, 2 & 3) we can deduce many others:
- If two rows of a matrix are equal, its determinant is zero.
This is because of property 2, the exchange rule.
On the one hand, exchanging the two identical rows does not change the determinant.
On the other hand, exchanging the two rows changes the sign of the determinant. Therefore the
determinant must be 0.
- If 𝑖 ≠ 𝑗, subtracting k times row 𝑖 from row 𝑗 doesn’t change the Determinant.
In two dimensions, this argument looks like;
We have ቚ
ଶ
ଶ
ଵ
ቚ property 3(b)
ቚ property 3(a)
ቚ − 𝑘 0 property 4
- If A has a row that is all zeros, then 𝑑𝑒𝑡 𝐴 = 0.
We get this from property 3 (a) by letting 𝑘 = 0.
- The determinant of a triangular matrix is the product of the diagonal entries (pivots) d1, d2, ...,
dn.
This is very useful. It is true that the determinant of a product equals the product of the
determinants.
Although the determinant of a sum does not equal the sum of the determinants.
i.e, 𝑑𝑒𝑡
௧
ቚ = ad − bc = ቚ
- If
= 0, then
