Matrix Operations | Mathematics (Maths) Class 12 - JEE PDF Download

Algebra of matrix involves the operation of matrices, such as Addition, subtraction, multiplication etc.

Let us understand the operation of the matrix in a much better way-

1. Addition of Matrices : 

Let A and B be two matrices of same order (i.e. comparable matrices). Then A + B is defined to be

2. Substraction of Matrices :

 Let A & B be two matrices of same order. Then A – B is defined as A + (–B) where – B is (–1) B.

3. Multiplication of Matrix By Scalar : 

Let λ be a scalar (real or complex number) & A = [a_ij]_{m × n} be a matrix. Thus the product λA is defined as λA = [b_ij]_{m × n} where b_ij = λa_ij for all i & j.
Note : If A is a scalar matrix, then A = λI, where λ is the diagonal element.

Properties of Addition & Scalar Multiplication : 

Consider all matrices of order m × n, whose elements are from a set F (F denote Q, R or C).

Let M_{m× n} (F) denote the set of all such matrices. Then

4. Multiplication of Matrices : 

Let A and B be two matrices such that the number of columns of A is same as number of rows of B. i.e., A =   where   which is the dot product of i^th row vector of A and j^th column vector of B.
Note :

1. The product AB is defined iff the number of columns of A is equal to the number of rows of B. A is called as premultiplier & B is called as post multiplier. AB is defined  BA is defined.
2. In general AB    BA, even when both the products are defined.
3. A(BC) = (AB) C, whenever it is defined.

Properties of Matrix Multiplication : 

Consider all square matrices of order ‘n’. Let M_n (F) denote the set of all square matrices of order n, (where F is Q, R or C). Then
(a) A, B ∈ M_n (F) ⇒ AB ∈ M_n(F)

(b) In general AB ≠ BA

(c) (AB) C = A(BC)

(d) I_n, the identity matrix of order n, is the multiplicative identity. AI_n = A = I_nA

(e) For every non singular matrix A(i.e., |A| ≠ 0) of Mn (F) there exist a unique (particular) matrix B ∈ Mn (F) so that AB = I_n = BA. In this case we say that A & B are multiplicative inverse of one another. In notations, we write B = A^-1 or A = B^-1.

(f) If λ is a scalar (λA) B = λ(AB) = A(λB).

Note :

1. Let A = [a_ij]_{m × n}. Then AI_n = A & I_m A = A, where I_n & I_m are identity matrices of order n & m respectively.

2. For a square matrix A, A² denotes AA, A³ denotes AAA etc.

Solved Examples:

Ex.1 For the following pairs of matrices, determine the sum and difference, if they exist.
(a)  

(b) 

Sol. (a) Matrices A and B are 2 × 3 and confirmable for addition and subtraction.

(b) Matrix A is 2 × 2, and B is 2 × 3. Since A and B are not the same size, they are not confirmable for addition or subtraction.
Ex.2 Find the additive inverse of the matrix A =
Sol. The additive inverse of the 3 × 4 matrix A is the 3 × 4 matrix each of whose elements is the negative of the corresponding element of A. Therefore if we denote the additive inverse of A by – A, we have . Obviously A + (–A) = (–A) + A = O, where O is the null matrix of the type 3 × 4.

Ex.3 If    find the matrix D such that A + B – D = 0.

Sol.  We have A + B – D = 0  ⇒  (A + B) + (-D) = 0  ⇒  A + B = (-D) = D  

Ex.4 If   verify that 3(A + B) = 3A + 3B.

Sol.

∴ 3 (A + B) = 3A + 3B, i.e. the scalar multiplication of matrices distributes over the addition of matrices.
 

Ex.5 The set of natural numbers N is partitioned into arrays of rows and columns in the form of matrices as 

and so on. Find the sum of the elements of the diagonal in M_n.

Sol. Let M_n = (a_ij) where i, j = 1, 2, 3,.........,n.
We first find out a₁₁ for the nth matrix; which is the nth term in the series ;         1, 2, 6,......
Let S = 1 + 2 + 6 + 15 +..... + T_{n – 1} + T_n.

Again writing        S = 1 + 2 + 6 +.... + T_{n – 1} + T_n

⇒ 0 = 1 + 1 + 4 + 9 +..... + (T_n – T_{n – 1}) – T_n ⇒ T_n = 1 + (1 + 4 + 9 +....... upto (n – 1) terms)

= 1 + (1 ² + 2² + 3² + 4² +..... + (n – 1)²)  

Ex.6  

Sol.

The matrix AB is of the type 3 × 3 and the matrix BA is also of the type 3 × 3. But the corresponding elements of these matrices are not equal. Hence AB≠ BA.

Ex.7 Show that for all values of p, q, r, s the matrices,  

Sol.   

 for all values of p, q, r,s. Hence PQ = QP, for all values of p, q, r, s.

Ex.8   where k is any positive integer.
Sol. We shall prove the result by induction on k.

We shall prove the result by induction on k.

We have   Thus the result is true when k = 1.

Now suppose that the result is true for any positive integer k.

 where k is any positive integer.

Now we shall show that the result is true for k + 1 if it is true for k. We have

Thus the result is true for k + 1 if it is true for k. But it is true for k = 1. Hence by induction it is true for all positive integral value of k.

Ex.9     where I is the two rowed unit matrix n is a positive integer.

Sol.

  = 0  ⇒  A³ = A² . A = 0  ⇒  A² = A³ = A⁴ =...... Aⁿ = 0

Now by binomial theorem

(a I + b A)ⁿ = (a I)ⁿ + ⁿC₁(a I)^{n – 1} b A + ⁿC₂ (a I)^{n – 2} (b A)² +..... + ⁿC_n (b A)ⁿ

= aⁿ I + ⁿC₁ a^{n – 1} b I A + ⁿC₂ an – 2 b² I A² +...... + ⁿC_n b_n A_n

= aⁿ I + n a^{n – 1} b A + 0......

Ex.10 If    then find the value of (n + a).
Sol.  Consider 

Hence n = 9 and 2007 =  

⇒ 2007 = 9a + 32 · 9 = 9(a + 32)

⇒   a + 32 = 223  ⇒  a = 191

hence a + n = 200

Ex.11 Find the matrices of transformations T₁T₂ and T₂T₁, when T₁ is rotation through an angle 60º and T₂ is the reflection in the y–axis. Also verify that T₁T₂ T₂T₁.

Sol.

...(1)

It is clear from (1) and (2), T₁T₂ ≠ T₂T₁

Ex.12 Find the possible square roots of the two rowed unit matrix I.

Sol.

  be square root of the matrix  =  

Since the above matrices are equal, therefore

a² + bc =1 ...(i)

ab + bd = 0 ...(ii)

ac + cd = 0 ,,,(iii)

cb + d² = 0 ....(iv)

must hold simultaneously.
If a + d = 0, the above four equations hold simultaneously if d = –a and a² + bc =1

Hence one possible square root of I is

 where α, β, γ are any three numbers related by the condition α² + βγ = 1.

If a + d ≠ 0, the above four equations hold simultaneously if b = 0, c = 0, a = 1, d = 1 or if b = 0, c = 0, a = –1, d = –1. Hence    i.e.  ± I are other possible square roots of I.

Ex.13  then prove that 

Sol.

...(i)

...(ii)

 ...(iii)

Similarly it can be shown that A⁴ = 2³ x⁴ E, A⁵ = 2⁴ x⁵ E ...

D. FURTHER TYPES OF MATRICES

(a) Nilpotent matrix : A square matrix A is said to be nilpotent (of order 2) if, A² = O.
A square matrix is said to be nilpotent of order p, if p is the least positive integer such that A^p = O

(b) Idempotent matrix : A square matrix A is said to be idempotent if, A² = A. eg.  is an idempotent matrix.

(c) Involutory matrix : A square matrix A is said to be involutory if A² = I,I being the identity matrix. eg. A =  is an involutory matrix.

(d) Orthogonal matrix : A square matrix A is said to be an orthogonal matrix if A'A = I = A'A

(e) Unitary matrix : A square matrix A is said to be unitary if  is the complex conjugate of A.

Ex.14 Find the number of idempotent diagonal matrices of order n.

Sol. Let A = diag (d₁, d₂,....., d_n) be any diagonal matrix of order n.

now A² = A . A =   

But A is idempotent, so A² = A and hence corresponding elements of A² and A should be equal

∴    or d₁ = 0, 1; d₂ = 0, 1;.........;d_n = 0, 1

⇒ each of d₁, d₂ ......, dn can be filled by 0 or 1 in two ways.

⇒ Total number of ways of selecting d₁, d₂,......., d_n = 2ⁿ

Hence total number of such matrices = 2ⁿ.

Ex.15 Show that the matrix A =  is nilpotent and find its index.

Sol. 

Thus 3 is the least positive integer such that A3 = 0. Hence the matrix A is nilpotent of index 3.

Ex.16 If AB = A and BA = B then B'A' = A' and A'B' = B' and hence prove that A' and B' are idempotent.

Sol. We have AB = A ⇒ (AB)' = A' ⇒ B'A' = A'. Also BA = B ⇒ (BA)' = B' ⇒ A'B' = B'.

Now A' is idempotent if A'2 = A'. We have A'2 =A'A' = A' (B'A') = (A'B') A' = B'A' =A'.

∴ A' is idempotent.

Again B'² = B'B' = B' (A'B') = (B'A') B' = A'B' = B'. ∴ B' is idempotent.

E. TRANSPOSE OF MATRIX

Let A = [a_ij]_{m × n}. Then the transpose of A is denoted by A'(or A^T) and is defined as A' = [b_ij]_{n × m} where b_ij = a_ji  for all i & j

i.e. A' is obtained by rewriting all the rows of A as columns (or by rewriting all the columns of A as rows).

(i) For any matrix A = [aij]_{m × n}, (A')' = A

(ii) Let λ be a scalar & A be a matrix. Then (λA)' = λA'

(iii) (A + B)' = A' + B' & (A - B)' = A' - B' for two comparable matrices A and B.

(iv) (A₁ ± A₂ ±.... ± A_n)' = A₁' ± A₂' ±..... ± A_n', where A_j are comparable.

(v) Let A = [a_ij]_{m × p} & B = [b_ij]_{p × n}, then (AB)' = B'A'

(vi) (A₁ A₂ ......... A_n)' = An'. A_n-1'..................± A₂'. A₁', provided the product is defined.

(vii) Symmetric & Skew–Symmetric Matrix :  A square matrix A is said to be symmetric if A' = A 

i.e. Let A = [a_ij]n. A is symmetric iff a_ij = a_ij for all i & j.
A square matrix A is said to be skew–symmetric if A' = - A

i.e. Let A = [a_ij]_n. A is skew–symmetric iff a_ij = –a_ji for all i & j.

 is a symmetric matrix 

   is a skew–symmetric matrix.

 Note :

1. In skew–symmetric matrix all the diagonal elements are zero. (a_ij = - a_ij  ⇒  a_ij = 0)

2. For any square matrix A, A + A' is symmetric & A -A' is skew - symmetric.

3. Every square matrix can be uniquely expressed as a sum of two square matrices of which one is symmetric and the other is skew–symmetric.