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**C. OPERATIONS ON MATRICES**

**(i) Equality of Matrices :** Two matrices A ad B are said to be equal if they are comparable and all the corresponding elements are equal.

Let A = [a_{ij}]_{m Ã— n} & B = [b_{ij}] _{p Ã— q}. A = B if (i) m = p, n = q (ii) a_{ij }= b_{ij} for i & j

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

**(iii)Substraction of Matrices** : Let A & B be two matrices of same order. Then A â€“ B is defined as A + (â€“B) where â€“ B is (â€“1) B.

**(iv)Multiplication of Matrix By Scalar :** Let Î» be a scalar (real or complex number) & A = [a_{ij}]_{m Ã— n }be a matrix. Thus the product Î»A is defined as Î»A = [b_{ij}]_{m Ã— n} where b_{ij} = Î»a_{ij }for all i & j.**Note : **If A is a scalar matrix, then A = Î»I, where Î» is the diagonal element.

**(v) Properties of Addition & Scalar Multiplication :** Consider all matrices of order m Ã— n, whose elements are from a set F (F denote Q, R or C).

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

**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 Ã— 3 and confirmable for addition and subtraction.

(b) Matrix A is 2 Ã— 2, and B is 2 Ã— 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 Ã— 4 matrix A is the 3 Ã— 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 â€“ A, we have . Obviously A + (â€“A) = (â€“A) + A = O, where O is the null matrix of the type 3 Ã— 4.

**Ex.3 If ** **find the matrix D such that A + B â€“ D = 0.**

**Sol. **We have A + B â€“ D = 0 â‡’ (A + B) + (-D) = 0 â‡’ A + B = (-D) = D

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

**Sol.**

âˆ´ 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 a_{11} for the nth matrix; which is the nth term in the series ; 1, 2, 6,......

Let S = 1 + 2 + 6 + 15 +..... + T_{n â€“ 1} + T_{n}.

Again writing S = 1 + 2 + 6 +.... + T_{n â€“ 1} + T_{n}

â‡’ 0 = 1 + 1 + 4 + 9 +..... + (T_{n }â€“ T_{n â€“ 1}) â€“ T_{n} â‡’ T_{n} = 1 + (1 + 4 + 9 +....... upto (n â€“ 1) terms)

= 1 + (1^{ 2 }+ 2^{2} + 3^{2} + 4^{2} +..... + (n â€“ 1)^{2})

**(vi) 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.

**(vii) Properties of Matrix Multiplication :** Consider all square matrices of order â€˜nâ€™. 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 âˆˆ M_{n} (F) â‡’ AB âˆˆ M_{n}(F)

(b) In general AB â‰ BA

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

(d) I_{n}, the identity matrix of order n, is the multiplicative identity. AI_{n} = A = I_{n}A

(e) For every non singular matrix A(i.e., |A| â‰ 0) of Mn (F) there exist a unique (particular) matrix B âˆˆ 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 Î» is a scalar (Î»A) B = Î»(AB) = A(Î»B).

**Note :**

1. Let A = [a_{ij}]_{m Ã— 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, A^{2} denotes AA, A^{3} denotes AAA etc.

**Ex.6 **

**Sol.**

The matrix AB is of the type 3 Ã— 3 and the matrix BA is also of the type 3 Ã— 3. But the corresponding elements of these matrices are not equal. Hence ABâ‰ 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 â‡’ A^{3} = A^{2} . A = 0 â‡’ A^{2} = A^{3} = A^{4} =...... A^{n} = 0

Now by binomial theorem

(a I + b A)^{n} = (a I)^{n} + ^{n}C_{1}(a I)^{n â€“ 1} b A + ^{n}C_{2} (a I)^{n â€“ 2} (b A)^{2} +..... + ^{n}C_{n }(b A)^{n}

= a^{n} I + ^{n}C_{1} a^{n â€“ 1 }b I A + ^{n}C_{2 }an â€“ 2 b^{2} I A^{2} +...... + ^{n}C_{n }b_{n }A_{n}

= a^{n} I + n a^{n â€“ 1 }b A + 0......

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

**Sol.** Consider

Hence n = 9 and 2007 =

â‡’ 2007 = 9a + 32 Â· 9 = 9(a + 32)

â‡’ a + 32 = 223 â‡’ a = 191

hence a + n = 200

**Ex.11 Find the matrices of transformations T _{1}T_{2} and T_{2}T_{1}, when T_{1} is rotation through an angle 60Âº and T_{2} is the reflection in the yâ€“axis. Also verify that T_{1}T_{2} **

**Sol.**

...(1)

It is clear from (1) and (2), T_{1}T_{2} â‰ T_{2}T_{1}

**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

a^{2} + bc =1 ...(i)

ab + bd = 0 ...(ii)

ac + cd = 0 ,,,(iii)

cb + d^{2} = 0 ....(iv)

must hold simultaneously.

If a + d = 0, the above four equations hold simultaneously if d = â€“a and a^{2} + bc =1

Hence one possible square root of I is

where Î±, Î², Î³ are any three numbers related by the condition Î±^{2} + Î²Î³ = 1.

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

**Ex.13 **** then prove that **

**Sol.**

...(i)

...(ii)

...(iii)

Similarly it can be shown that A^{4} = 2^{3} x^{4} E, A^{5 }= 2^{4} x^{5} E ...

**D. FURTHER TYPES OF MATRICES**

**(a) Nilpotent matrix :** A square matrix A is said to be nilpotent (of order 2) if, A^{2} = 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, A^{2} = A. eg. is an idempotent matrix.

**(c) Involutory matrix :** A square matrix A is said to be involutory if A^{2} = 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 (d_{1}, d_{2},....., d_{n}) be any diagonal matrix of order n.

now A^{2} = A . A =

But A is idempotent, so A^{2} = A and hence corresponding elements of A^{2} and A should be equal

âˆ´ or d_{1} = 0, 1; d_{2} = 0, 1;.........;d_{n} = 0, 1

â‡’ each of d_{1}, d_{2} ......, dn can be filled by 0 or 1 in two ways.

â‡’ Total number of ways of selecting d_{1}, d_{2},......., d_{n} = 2^{n}

Hence total number of such matrices = 2^{n}.

**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 â‡’ (AB)' = A' â‡’ B'A' = A'. Also BA = B â‡’ (BA)' = B' â‡’ 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'.

âˆ´ A' is idempotent.

Again B'^{2} = B'B' = B' (A'B') = (B'A') B' = A'B' = B'. âˆ´ B' is idempotent.

**E. TRANSPOSE OF MATRIX**

Let A = [a_{ij}]_{m Ã— n}. Then the transpose of A is denoted by A'(or A^{T}) and is defined as A' = [b_{ij}]_{n Ã— 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 Ã— n}, (A')' = A

(ii) Let Î» be a scalar & A be a matrix. Then (Î»A)' = Î»A'

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

(iv) (A_{1} Â± A_{2} Â±.... Â± A_{n})' = A_{1}' Â± A_{2}' Â±..... Â± A_{n}', where A_{j} are comparable.

(v) Let A = [a_{ij}]_{m Ã— p} & B = [b_{ij}]_{p Ã— n}, then (AB)' = B'A'

(vi) (A_{1} A_{2 }......... A_{n})' = An'. A_{n-1}'..................Â± A_{2}'. A_{1}', provided the product is defined.

**(vii) Symmetric & Skewâ€“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 skewâ€“symmetric if A' = - A

i.e. Let A = [a_{ij}]_{n}. A is skewâ€“symmetric iff a_{ij} = â€“a_{ji} for all i & j.

is a symmetric matrix

is a skewâ€“symmetric matrix.

**Note :**

1. In skewâ€“symmetric matrix all the diagonal elements are zero. (a_{ij} = - a_{ij} â‡’ 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 skewâ€“symmetric.

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