## NCERT Notes for Class 12 Mathematics

Chapter 3: Matrices

A matrix is a rectangular arrangement of numbers (real or complex) which may be represented
as

matrix is enclosed by [ ] or ( ) or | | | |
Compact form the above matrix is represented by [a_{ij}]_{m x n} or A = [a_{ij}].

**1. **Element of a Matrix The numbers a_{11}, a_{12} … etc., in the above matrix are known as the
element of the matrix, generally represented as a_{ij} , which denotes element in ith row and
jth column.

**2.** Order of a Matrix In above matrix has m rows and n columns, then A is of order m x n.

### Types of Matrices

**1.** Row Matrix A matrix having only one row and any number of columns is called a row
matrix.

**2.** Column Matrix A matrix having only one column and any number of rows is called
column matrix.

**3. **Rectangular Matrix A matrix of order m x n, such that m ≠ n, is called rectangular
matrix.

**4.** Horizontal Matrix A matrix in which the number of rows is less than the number of
columns, is called a horizontal matrix.

**5.** Vertical Matrix A matrix in which the number of rows is greater than the number of
columns, is called a vertical matrix.

**6.** 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

**7.** Square Matrix A matrix of order m x n, such that m = n, is called square matrix.

**8.** 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}]

**9. **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

**10.** 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.

**11.** Upper Triangular Matrix A square matrix A = a[_{ij}]_{n x n} is called a upper triangular matrix, if a[_{ij}], = 0, ∀ i > j.

**12.** Lower Triangular Matrix A square matrix A = a[_{ij}]_{n x n} is called a lower triangular matrix, if a[_{ij}], = 0, ∀ i < j.

**13.** Submatrix A matrix which is obtained from a given matrix by deleting any number of rows or columns or both is called a submatrix of the given matrix.

**14.** 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.

**15.** Principal Diagonal of a Matrix In a square matrix, the diagonal from the first element of

the first row to the last element of the last row is called the principal diagonal of a
matrix.

**16.** 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

1. 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} + b_{ij}]_{m x n}

##### Properties of Addition of Matrices

If A, B and C are three matrices of order m x n, then

**1. Commutative Law** A + B = B + A

**2. Associative Law** (A + B) + C = A + (B + C)

**3. 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

**4. Existence of Additive Inverse** If A is a square matrix, then the matrix (- A) is called
additive inverse, if A + ( – A) = 0 = (- A) + A

**5. Cancellation Law**

A + B = A + C ⇒ B = C (left cancellation law)

B + A = C + A ⇒ B = C (right cancellation law)

#### 2. 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 }– b_{ij}]_{n x n},
where A = [a_{ij}]_{m x n}, B = [b_{ij}]_{m x n}

#### 3. 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

**1.** k(A + B) = kA + kB

**2.** (k_{1} + k_{2})A = k_{1}A + k_{2}A

**3.** k_{1}k_{2}A = k_{1}(k_{2}A) = k_{2}(k_{1}A)

**4.** (- k)A = – (kA) = k( – A)

#### 4. 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, is given by

where c_{ij }is the element of matrix C and C = AB

### Properties of Multiplication of Matrices

**1.** Commutative Law Generally AB ≠ BA

**2. **Associative Law (AB)C = A(BC)

**3.** Existence of multiplicative Identity A.I = A = I.A,
I is called multiplicative Identity.

**4. **Distributive Law A(B + C) = AB + AC

**5.** Cancellation Law If A is non-singular matrix, then

AB = AC ⇒ B = C (left cancellation law)

BA = CA ⇒B = C (right cancellation law)

**6.** 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).

(_{ii}i) 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.

##### Positive Integral Powers of a Square Matrix

Let A be a square matrix. Then, we can define

**1. **A_{n + 1} = A_{n}. A, where n ∈ N.

**2.** A^{m}. A_{n} = A^{m + n}

**3.** (Am)_{n} = A^{mn}, ∀ m, n ∈ N

##### Matrix Polynomial

Let f(x)= a^{0}x^{n} + a1x^{n – 1} ^{-1} + a2x^{n – 2} + … + an. Then f(A)= a_{0}A^{n} + a_{1}A^{n – 2} + … + a_{n}I_{n} is called the matrix polynomial.

##### Transpose of a Matrix

Let A = [a_{ij}]_{m x n}, be a matrix of order m x n. Then, the n x m matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by ’or A^{T}.

A’ = AT = [a_{ij}]_{n x m}

##### Properties of Transpose

**1. **(A’)’ = A

**2.** (A + B)’ = A’ + B’

**3.** (AB)’ = B’A’

**4. **(KA)’ = kA’

**5.** (A^{N})’ = (A’)^{N}

**6.** (ABC)’ = C’ B’ A’

#### Symmetric and Skew-Symmetric Matrices

**1.** A square matrix A = [a_{ij}]<<, is said to be symmetric, if A’ = A.

i.e., a_{ij} = a_{ji} , ∀i and j.

**2.** A square matrix A is said to be skew-symmetric matrices, if i.e., a_{ij} = — a_{ji}, di and j

#### Properties of Symmetric and Skew-Symmetric Matrices

**1.** Elements of principal diagonals of a skew-symmetric matrix are all zero. i.e., a_{ii} = —
a_{ii} 2< = 0 or a_{ii} = 0, for all values of i.

**2.** If A is a square matrix, then

(a) A + A’ is symmetric.

(b) A — A’ is skew-symmetric matrix.

**3.** If A and B are two symmetric (or skew-symmetric) matrices of same order, then A + B
is also symmetric (or skew-symmetric).

**4.** If A is symmetric (or skew-symmetric), then kA (k is a scalar) is also symmetric for
skew-symmetric matrix.

**5. **If A and B are symmetric matrices of the same order, then the product AB is symmetric,
iff BA = AB.

**6. **Every square matrix can be expressed uniquely as the sum of a symmetric and a skewsymmetric
matrix.

**7. **The matrix B’ AB is symmetric or skew-symmetric according as A is symmetric or
skew-symmetric matrix.

**8. **All positive integral powers of a symmetric matrix are symmetric.

**9.** All positive odd integral powers of a skew-symmetric matrix are skew-symmetric and
positive even integral powers of a skew-symmetric are symmetric matrix.

**10.** If A and B are symmetric matrices of the same order, then

(a) AB – BA is a skew-symmetric and

(b) AB + BA is symmetric.

**11.** For a square matrix A, AA’ and A’ A are symmetric matrix.

##### 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

**1.** Trace (A ± B)= Trace (A) ± Trace (B)

**2. **Trace (kA)= k Trace (A)

**3. **Trace (A’ ) = Trace (A)

**4. **Trace (I_{n})= n

**5.** Trace (0) = 0

**6.** Trace (AB) ≠ Trace (A) x Trace (B)

**7.** Trace (AA’) ≥ 0

##### Conjugate of a Matrix

If A is a matrix of order m x n, then

#### Transpose Conjugate of a Matrix

The transpose of the conjugate of a matrix A is called transpose conjugate of A and is denoted
by A^{0} or A^{*.
i.e., (A’) = A‘ = A0 or A*}

##### Properties of Transpose Conjugate of a Matrix

(i) (A^{*)* = A
(ii) (A + B)* = A* + B*
(iii) (kA)* = kA*
(iv) (AB)* = B*A*
(V) (An)* = (A*)n}

##### Some Special Types of Matrices

###### 1. Orthogonal Matrix

A square matrix of order n is said to be orthogonal, if AA’ = I_{n} = A’A Properties of Orthogonal
Matrix

(i) If A is orthogonal matrix, then A’ is also orthogonal matrix.

(ii) For any two orthogonal matrices A and B, AB and BA is also an orthogonal matrix.

(iii) If A is an orthogonal matrix, A^{-1} is also orthogonal matrix.

###### 2. ldempotent Matrix

A square matrix A is said to be idempotent, if A^{2} = A.

##### Properties of Idempotent Matrix

(i) If A and B are two idempotent matrices, then

• AB is idempotent, if AB = BA.

• A + B is an idempotent matrix, iff
AB = BA = 0

• AB = A and BA = B, then A^{2} = A, B^{2} = B

(ii)

• If A is an idempotent matrix and A + B = I, then B is an idempotent and AB = BA= 0.

• Diagonal (1, 1, 1, …,1) is an idempotent matrix.

• If I_{1}, I_{2} and I_{3} are direction cosines, then

is an idempotent as |Δ|^{2} = 1.

A square matrix A is said to be involutory, if A^{2} = I

###### 4. Nilpotent Matrix

A square matrix A is said to be nilpotent matrix, if there exists a positive integer m such that
A^{2} = 0. If m is the least positive integer such that A^{m} = 0, then m is called the index of the
nilpotent matrix A.

###### 5. Unitary Matrix

A square matrix A is said to be unitary, if A‘A = I

#### Hermitian Matrix

A square matrix A is said to be hermitian matrix, if A = A^{*} or = a_{ij}, for a_{ji} only.

###### Properties of Hermitian Matrix

**1.** If A is hermitian matrix, then kA is also hermitian matrix for any non-zero real number k.

**2.** If A and B are hermitian matrices of same order, then λ_{λ}A + λB, also hermitian for any
non-zero real number λ_{λ}, and λ.

**3. **If A is any square matrix, then AA^{*} and A^{*} A are also hermitian.

**4.** If A and B are hermitian, then AB is also hermitian, iff AB = BA

**5.** If A is a hermitian matrix, then A is also hermitian.

**6. **If A and B are hermitian matrix of same order, then AB + BA is also hermitian.

**7.** If A is a square matrix, then A + A^{*} is also hermitian,

**8.** Any square matrix can be uniquely expressed as A + iB, where A and B are hermitian matrices.

#### Skew-Hermitian Matrix

A square matrix A is said to be skew-hermitian if A^{*} = – A or a_{ji} for every i and j.

#### Properties of Skew-Hermitian Matrix

**1.** If A is skew-hermitian matrix, then kA is skew-hermitian matrix, where k is any nonzero
real number.

**2.** If A and B are skew-hermitian matrix of same order, then λ_{λ}A + λ_{2}B is also skewhermitian
for any real number λ_{λ} and λ_{2}.

**3.** If A and B are hermitian matrices of same order, then AB — BA is skew-hermitian.

**4.** If A is any square matrix, then A — A^{*} is a skew-hermitian matrix.

**5.** Every square matrix can be uniquely expressed as the sum of a hermitian and a skewhermitian
matrices.

**6. **If A is a skew-hermitian matrix, then A is a hermitian matrix.

**7. **If A is a skew-hermitian matrix, then A is also skew-hermitian matrix.

#### Adjoint of a Square Matrix

Let A[a_{ij}]m x n be a square matrix of order n and let C_{ij} be the cofactor of a_{ij} in the determinant
|A| , then the adjoint of A, denoted by adj (A), is defined as the transpose of the matrix, formed
by the cofactors of the matrix.

#### Properties of Adjoint of a Square Matrix

If A and B are square matrices of order n, then

**1. **A (adj A) = (adj A) A = |A|I

**2.** adj (A’) = (adj A)’

**3.** adj (AB) = (adj B) (adj A)

**4.** adj (kA) = k^{n – 1}(adj A), k ∈ R

**5.** adj (A^{m}) = (adj A)^{m}

**6.** adj (adj A) = |A|^{n – 2} A, A is a non-singular matrix.

**7.** |adj A| =|A|^{n – 1} ,A is a non-singular matrix.

**8.** |adj (adj A)| =|A|^{(n – 1)2} A is a non-singular matrix.

**9.** Adjoint of a diagonal matrix is a diagonal matrix.

#### Inverse of a Square Matrix

Let A be a square matrix of order n, then a square matrix B, such that AB = BA = I, is called inverse of A, denoted by A^{-1}.

i.e.,

or AA^{-1} = A^{-1}A = 1

##### Properties of Inverse of a Square Matrix

**1. **Square matrix A is invertible if and only if |A| ≠ 0

**2. **(A^{-1})^{-1} = A

**3.** (A’)^{-1} = (A^{-1})’

**4.** (AB)^{-1} = B^{-1}A^{-1} In general (A_{1}A_{1}A_{1} … A_{n})^{-1} = A_{n} ^{-1}A_{n} – 1 ^{-1} … A_{3}^{-1}A2 ^{-1}A_{1} ^{-1}

**5.** If a non-singular square matrix A is symmetric, then A^{-1} is also symmetric.

**6.** |A^{-1}| = |A|^{-1}

7. AA^{-1} = A^{-1}A = I

**8.** (A^{k})^{-1} = (A^{-1})A^{k} k ∈ N

#### Elementary Transformation

Any one of the following operations on a matrix is called an elementary transformation.

**1.** Interchanging any two rows (or columns), denoted by R_{i}←→Rj or C_{i}←→C_{j}

**2.** Multiplication of the element of any row (or column) by a non-zero quantity and denoted
by
R_{i} → kR_{i} or C_{i} → kC_{j}

3. Addition of constant multiple of the elements of any row to the corresponding elementof any other row, denoted by
R_{i} → R_{i} + kRj or C_{i} → C_{i} + kC_{j}

##### Equivalent Matrix

• Two matrices A and B are said to be equivalent, if one can be obtained from the other by
a sequence of elementary transformation.

• The symbol≈ is used for equivalence.

##### Rank of a Matrix

A positive integer r is said to be the rank of a non-zero matrix A, if

**1.** there exists at least one minor in A of order r which is not zero.

**2.** every minor in A of order greater than r is zero, rank of a matrix A is denoted by ρ(A) =
r.

##### Properties of Rank of a Matri**x**

** 1.** The rank of a null matrix is zero ie, ρ(0) = 0

**2.** If In is an identity matrix of order n, then ρ(I_{n}) = n.

**3.** (a) If a matrix A does’t possess any minor of order r, then ρ(A) ≥ r.

(b) If at least one minor of order r of the matrix is not equal to zero, then ρ(A) ≤ r.

**4.** If every (r + 1)th order minor of A is zero, then any higher order – minor will also be
zero.

**5.** If A is of order n, then for a non-singular matrix A, ρ(A) = n

6. ρ(A’)= ρ(A)

**7.** ρ(A^{*}) = ρ(A)

**8.** ρ(A + B) &LE; ρ(A) + ρ(B)

9. If A and B are two matrices such that the product AB is defined, then rank (AB) cannot
exceed the rank of the either matrix.

**10.** If A and B are square matrix of same order and ρ(A) = ρ(B) = n, then p(AB)= n

**11.** Every skew-symmetric matrix,of odd order has rank less than its order.

**12.** Elementary operations do not change the rank of a matrix.

##### Echelon Form of a Matrix

A non-zero matrix A is said to be in Echelon form, if A satisfies the following conditions

**1.** All the non-zero rows of A, if any precede the zero rows.

**2.** The number of zeros preceding the first non-zero element in a row is less than the
number of such zeros in the successive row.

**3.** The first non-zero element in a row is unity.

**4.** The number of non-zero rows of a matrix given in the Echelon form is its rank.

##### Homogeneous and Non-Homogeneous System of Linear Equations

A system of equations AX = B, is called a homogeneous system if B = 0 and if B ≠ 0, then it is
called a non-homogeneous system of equations.

##### Solution of System of Linear Equations

The values of the variables satisfying all the linear equations in the system, is called solution of system of linear equations.

##### 1 . Solution of System of Equations by Matrix Method

###### (i) Non-Homogeneous System of Equations

Let AX = B be a system of n linear equations in
n variables.

• If |A| ≠ 0, then the system of equations is consistent and has a unique solution given by X = A^{-1}B.

• If |A| = 0 and (adj A)B = 0, then the system of equations is consistent and has infinitely many solutions.

• If |A| = 0 and (adj A) B ≠ 0, then the system of equations is inconsistent i.e., having no solution

###### (ii) Homogeneous System of Equations

Let AX = 0 is a system of n linear equations in n variables.

• If I |A| ≠ 0, then it has only solution X = 0, is called the trivial solution.

• If I |A| = 0, then the system has infinitely many solutions, called non-trivial solution.

##### 2. Solution of System of Equations by Rank Method

###### (i) Non-Homogeneous System of Equations

Let AX = B, be a system of n linear equations in
n variables, then

**• Step I** Write the augmented matrix [A:B]

**• Step II **Reduce the augmented matrix to Echelon form using elementary owtransformation.

**• Step III **Determine the rank of coefficient matrix A and augmented matrix [A:B] by
counting the number of non-zero rows in A and [A:B].

##### Important Results

**1.** If ρ(A) ≠ ρ(AB), then the system of equations is inconsistent.

**2.** If ρ(A) =ρ(AB) = the number of unknowns, then the system of equations is consistent
and has a unique solution.

3. If ρ(A) = ρ(AB) < the number of unknowns, then the system of equations is consistent
and has infinitely many solutions.

(ii) Homogeneous System of Equations

• If AX = 0, be a homogeneous system of linear equations then, If ρ(A) = number of
unknown, then AX = 0, have a non-trivial solution, i.e., X = 0.

• If ρ(A) < number of unknowns, then AX = 0, have a non-trivial solution, with infinitely
many solutions.