Summary of Class 12 Maths Matrices
Definition of matrix: A matrix is a rectangular array of numbers
represented as rows and columns. 
Where m represents the number of rows and n represents the
number of columns.
The number aij is an element lying in the ith
row and jth column. We usually write A = (aij).
Order of a matrix: The order of a matrix is represented by
the number of rows and the number of columns in a matrix. A matrix of order m ×
n (read m-cross-n) has m rows and n columns.
Types of Matrices
1.
Row
Matrix: A matrix which has only one row, is
called a row matrix.
2. Column Matrix: A matrix which has only one column, is called a column
matrix.
3.
Square Matrix: A matrix in which
the number of rows (m) is equal to the number of columns (n), is called a
square matrix. In this case, we say that A is of order n.
4.
Null Matrix (Zero Matrix): The matrix in which
all the elements are zero, is called the zero matrix or null matrix. It is
denoted by O.
5.
Diagonal Matrix: A square matrix A =
[aij] in which all the non-diagonal elements are zero, is
called a diagonal matrix. That is, in a diagonal matrix, aij = 0, if
i is not equal to j.
6.
Identity Matrix: A diagonal matrix in
which all the diagonal elements are equal to 1, is called the identity matrix. It
is denoted by I.
7.
Scalar Matrix: A diagonal matrix in
which all the diagonal elements are equal, is known as a scalar matrix. Thus, a
scalar matrix is of the form CI where C is a scalar (real or complex number)
and I is the identity matrix.
Equality of Matrices
Two matrices A = [aij]m ×
n and B = [bij]m × n of the same order are said to
be equal, if the corresponding elements of both the matrices are equal, that
is, aij = bij for all i, j.
Addition of Matrices
If A = [aij]m × n
and B = [bij]m × n are two matrices of the same order m ×
n, then their sum is defined as A + B = [aij + bij]m
× n. The sum (A + B) is also a matrix of order m × n.
Difference (or Subtraction) of Matrices
If A = [aij]m × n
and B = [bij]m × n are two matrices of the same order m ×
n, then their difference is defined as A – B
= [aij – bij]m × n. The difference (A –
B) is also a matrix of order m × n.
Multiplication of a Matrix by a Scalar
When matrix A is multiplied by a scalar k,
then all the elements of A get multiplied by the scalar k. That is, kA = k[aij]m
× n = [k(aij)]m × n
Multiplication of Two Matrices
Multiplication of two matrices is defined,
if the number of columns of A equals the number of rows of B.
If A = [aij]m × n
and B = [bjk]n × r be two matrices, then the
multiplication of A and B is denoted by AB and defined as C = 
,
where C is a matrix of order m × r and C = AB.
Properties of Multiplication of Matrices
1. Matrix multiplication is not
commutative, i.e., AB is not equal to BA in general. If A is of order 3 × 2 and
B is of order 2 × 2, then AB is defined but BA is not even defined.
2. Matrix multiplication is associative,
that is, if A, B and C are three matrices of same order, then A(BC) = (AB)C.
3. Matrix multiplication is distributive
over addition. If A, B and C are three matrices of same order, then A(B + C) =
AB + AC
4. For every square matrix A, there
exists an identity matrix I of the same order such that A.I = I.A = A.
Transpose of a Matrix
The transpose of a matrix A, denoted by A’ or AT,
is obtained by interchanging its rows and columns. Thus,
if A = [aij] then A’ = [aji]. Note that, if matrix A has
order m × n, then its transpose A’ has order n × m.
The properties of transpose of a given matrix
are as follows.
· The transpose of (transpose of a matrix A) results in the original matrix. (A')' = A.
· The transpose of the product of a constant and a matrix is equal to the product of the constant and the transpose of the matrix. (kA)' = kA', where k is any constant.
· The transpose of the sum or difference of two matrices is equal to the sum or difference of the transpose of the individual matrices. (A ± B)' = A' ± B'
· The transpose of
the product of two matrices is equal to the product of their transpose in the
reverse order. That is, (AB)' = B'A'
Symmetric and Skew-symmetric Matrices
Symmetric Matrix: A square
matrix A is called a symmetric matrix, if A’ = A. Let A = [aij],
then A is symmetric, if [aij] = [aji] for
all i, j.
Skew-symmetric Matrix: A square matrix A is
called skew symmetric, if A’ = -A. Let A = [aij],
then A is skew symmetric, if [aji] = –[aij]
for all i, j.
Properties
of Symmetric and Skew-symmetric Matrices
(i) For any square matrix A with real number entries, A + A' is a symmetric matrix, and A – A' is a skew-symmetric matrix.
(ii)
Any square matrix can be expressed as the sum of a symmetric and a
skew-symmetric matrix. Let A be a square matrix, then we can write
A = 
(A + A’) + (A – A’)