The Matrices

What is matrices ?

Defnition: A set of mn numbers (real or imaginary) arranged in the form of a rectangular array of m rows and n columns is called an m × n (to be read as ‘m by n’ matrices).

An m×n matrix is usually written as

$A=\begin{bmatrix} a_{11} & a_{12} & a_{13} &. . . & a_{1n} \\a_{21} & a_{22} & a_{23} &. . .& a_{2n} \\ a_{31} & a_{32} & a_{33}& . . .& a_{3n}\\ \vdots&\vdots&\vdots& &\vdots\\a_{m1} & a_{m2} & a_{m3} &. . . & a_{mn} \end{bmatrix}$

A compact form the above matrix is represented by $A=[a_{ij}]_{m\times n}$ or $A = [a_{ij}]$ .

Some points related to Matrices

(1) The numbers $a_{11},a_{12} . . .$ etc known as the elements of the matrices A.

(2) The element $a_{ij}$ belongs to ith row and jth column and is called the (i,j)th element of the matrix $A=[a_{ij}]$ .

(3) In the element $A=[a_{ij}]$ the first subscript i, always denotes the number of rows and the second subscript j, denotes the number of columns.

Following are some examples of matrices:

(i) $A=\begin{bmatrix}2 & 5 \\35 & -2 \\1 & -5 \end{bmatrix}$ is a matrix having 3 rows and 2 columns and so it is a matrix of order 3 × 2 such that $a_{11}= 2,a_{12}= 5,a_{21} = 35,a_{22}= -2,a_{13}=1, a_{32} = -5$ .

(ii) $A=\begin{bmatrix}\sin x & \cos x \\ \cos x & -\sin x \end{bmatrix}$ is a matrix having 2 rows and 2 columns and so it is a matrix of order 2 × 2 such that $a_{11}= \sin x,a_{12}= \cos x,a_{21} =\cos x,a_{22}= -\sin x$ .

Types of matrices

Row Matrix:(What is Row Matrix)

A matrix having only one row is called a row-matrix or a row vector .

For example, $A =\begin{bmatrix}2 & 5 & 1 & 6 \end{bmatrix}$ is a row matrix of order 1 × 4.

Column Matrix:(What is Column Matrix)

A matrix having only one column is called a column-matrix or a column-vector.

For example, $A =\begin{bmatrix}2 \\ 5 \\ 1 \\ 6 \end{bmatrix}$ is a column matrix of order 4 × 1.

Square matrix:(What is Scalar Matrix)

A matrix in which the number of rows is equal to the number of columns,say n, is called a square matrix of order n.

For example, The matrix $A=\begin{bmatrix}2 & 5 & 6 \\3 & -2 & 1 \\1 & -5 & 7 \end{bmatrix}$ is a square matrix of order 3

Diagonal Matrix:(What is Diagonal matrix ?)

A square matrix $A = [a_{ij}]_{n\times n}$ is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero i.e.

$a_{ij} = 0 \text{ For all} i \neq j$

If a diagonal matrix of order 3 × 3 having a,b,c as diagonal elements is denoted by diag[a, b, c].

Example: The matrix $A=\begin{bmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3 \end{bmatrix}$ is a diagonal matrix, to be denoted by A = diag[1, 2, 3]

Scalar Matrix:( What is scalar matrix ?)

A squarte matrix $A = [a_{ij}]_{n\times n}$ is called a scalar matrix if

(i) $a_{ij} = 0 \text{ For all} i \neq j$ , and

(ii) $a_{ij} = C \text{ For all} i = j$ where C ≠ 0

In other words, a diagonal matrix in which all the diagonal elements are equal is called the scalar matrix.

Example: The matrices $A=\begin{bmatrix}2 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 2\end{bmatrix}$ and $B=\begin{bmatrix}3 & 0 \\0 & 3 \end{bmatrix}$ are scalar matrices of order 3 and 2 respectively.

Identity or Unit matrix:( What is Identity or Unit Matrix)

A squarte matrix $A = [a_{ij}]_{n\times n}$ is called an identity or unit matrix if

(i) $a_{ij} = 0 \text{ For all} i \neq j$ , and

(ii) $a_{ij} = 1 \text{ For all} i = j$ where C ≠ 0

In other words, a square matrix each of whose diagonal elements is unity and each of whose non-diagonal elements is equal zero is called an identity or Unit matrix.

Identity matrix of order n is denoted by $I_n$ .

Example: The matrices $A=\begin{bmatrix}2 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 2\end{bmatrix}$ and $B=\begin{bmatrix}3 & 0 \\0 & 3 \end{bmatrix}$ are Identity matrices of order 3 and 2 respectively.

NULL MATRIX: (What is Null Matrix)

A Matrix whose all elements are zero is called a Null Matrix or Zero Matrix.

Example: $A=\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}, B=\begin{bmatrix}0 & 0 & 0 \\0 & 0 & 0 \end{bmatrix}$ are null matrices of order 2×2 and 2×3 respectively.

UPPER TRIANGULAR MATRIX:

A square matrix $A = [a_{ij}]$ is called an Upper Triangular Matrix if $a_{ij} = 0$ For all i>j.

Thus, in an Upper Triangular Matrix, all elements below the main diagonal are zero.

Example: $A=\begin{bmatrix}1 & 2 & 3 \\0 & 4 & 5 \\0 & 0 & 6\end{bmatrix}$ is an Upper Triangular Matrix.

LOWER TRIANGULAR MATRIX:

A square matrix $A = [a_{ij}]$ is called an Upper Triangular Matrix if $a_{ij} = 0$ For all i<j.

Thus, in an Lower Triangular Matrix, all elements above the main diagonal are zero.

Example: $A=\begin{bmatrix}1 & 0 & 0 \\2 & 3 & 0 \\4 & 5 & 6\end{bmatrix}$ is an Lower Triangular Matrix.

Strictly Triangular Matrix:

A Triangular Matrix $A = [a_{ij}]$ is called a strictly triangular matrix $a_{ij} = 0$ for all i = 1, 2, . . . ,n.

Equality Of Matrices

Two matrices $A = [a_{ij}]_{m\times n} \text{ and} B = [b_{ij}]_{r\times s}$ are equal if

(i) m = r, i.e., the number of rows in A equal the number of rows in B.

(ii) n = s, i.e., The number of column in A equals the number of column in B.

(iii) $a_{ij} = b_{ij}$ for i = 1, 2, . . ., m and j = 1, 2, . . ., n

If two matrices A and B are equal, we write A = B, otherwise we write A ≠ B.

Example 1:- If $\begin{bmatrix} x - y & 2x + z \\2x - y & 3z + w \end{bmatrix} = \begin{bmatrix}-1 & 5 \\0 & 13\end{bmatrix}$ find x, y, z, w.

Solution: Since the corresponding elements of two equal matrices are equal, therefore

$\begin{bmatrix} x - y & 2x + z \\2x - y & 3z + w \end{bmatrix} = \begin{bmatrix}-1 & 5 \\0 & 13\end{bmatrix}$

⇒ x – y = -1, 2x – y = 0, 2x + z = 5, 3z + w = 13.

Solving the equation and we get

x = 1, y = 2, z = 3, anw w = 4.

Example 2:- Matrices $\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}$ and $\begin{bmatrix}0 & 0 & 0 \\0 & 0 & 0\end{bmatrix}$ are not equal, because their order are not same.

Algebra Of Matrices:

Let A, B be two matrices, each of order m×n. Then their sum A + B is a matrix of order m×n and is obtained by adding the corresponding elements of A and B.

Thus, if $A = [a_{ij}]_{m\times n}$ and $B = [b_{ij}]_{m\times n}$ are two matrices of the same order, their sum A + B is defined to be the matrix of order m×n such that

$(A + B)_{ij} = a_{ij} + b_{ij}$ for i = 1, 2, . . , m and j = 1, 2, . . ., n

NOTE:- The sum of two matrices is defined only when they are of the same order.

Example:- If $A = \begin{bmatrix} 1 & 2 & 3\\4 & 5 & 6 \end{bmatrix},B = \begin{bmatrix} 6 & 5 & 4\\3 & 2 & 1 \end{bmatrix}$ , then

$A + B = \begin{bmatrix} 1+ 6 & 2+ 5 & 3 + 4\\4 + 3 & 5+ 2 & 6 + 1\end{bmatrix}$

$= \begin{bmatrix} 7 & 7 & 7\\7 & 7 & 7 \end{bmatrix}$

Properties Of Matrix Addition

(i) Matrix addition is commutative

A + B = B + A

(ii) Matrix addition is associative

A + (B + C) = (A + B) + C

(iii) Existence of Identity

A + O = A = O + A

The Null matrix is the identity matrix for matrix addition and it is denoted by O

(iv) Existence of Inverse

A + (-A) = O = (-A) + A

Inverse of matrix A is -A

(v) Cancellation laws hold good in case of addition of matrices

A + B = A + C ⇒ B = C

and B + A = C + A ⇒ B = C

Multiplication of a matrix by a scalar(Scalar multiplication)

Let $A = [a_{ij}]$ be an m×n matrix and k be any number called a scalar. Then the matrix obtained by multiplication every element of A by k is called the scalar multiple of A by k and is denoted by kA. Thus

$kA = [ka_{ij}]_{m\times n}$

Example:- If $A = \begin{bmatrix} 1 & 2 & 5 \\-2 & 3 & 4 \\ 1 & 2 & -1\end{bmatrix},\text{then} 3A = \begin{bmatrix} 3 & 6 & 15 \\-6 & 9 & 12 \\ 3 & 6 & -3\end{bmatrix}$

Properties of scalar multiplication

If $A = [a_{ij}]_{m\times n}, B = [b_{ij}]_{m\times n}$ are two matrices and k, l are scalars, then

(i) k(A + B) = kA + kB

(ii) (k + l)A = kA + lA

(iii) (kl)A = k(lA) = l(kA)

(iv)(-k)A = -(kA) = k(-A)

(v) 1A = A

(vi) (-1)A = -A.

Subtraction Of Matrices(Definition)

For two matrices A and B of the same order, we define A – B = A + (-B).

Example:- If $A = \begin{bmatrix} -3 & 2 & 1\\1 & -4 & 7 \end{bmatrix} \text{and} B = \begin{bmatrix} 3 & 5 & -2\\-1 & 4 & -2 \end{bmatrix}$ , then

$A - B = \begin{bmatrix} -3 & 2 & 1\\1 & -4 & 7 \end{bmatrix} - \begin{bmatrix} 3 & 5 & -2\\-1 & 4 & -2 \end{bmatrix}$

$= \begin{bmatrix} -6 & -3 & 3\\2 & -8 & 9 \end{bmatrix}$

Multiplication Of Matrices

Two matrices A and B are conformable for the product AB if the number of column in A(Post multiplier) is same as the number of rows in B(Pre multiplier). Thus, if $A= [a_{ij}]_{m\times n} \text{and} B = [b_{ij}]_{n\times p}$ are two matrices of order m×n and n×p respectively, then their product AB is of order m×p

Let $A = \begin{bmatrix} a & b & c \end{bmatrix}$ matrix of order 1×3 and $B = \begin{bmatrix} p \\ q \\r\end{bmatrix}$ is a matrix of order 3×1

The column of matrix A is 3 and the row of matrix B is also 3, then the AB is defined

Now, $AB = \begin{bmatrix} a & b & c \end{bmatrix}.\begin{bmatrix} p \\ q \\r\end{bmatrix}$

= ap + bq + cr

NOTE:- If A and B are two matrices such that AB exists, then BA may or may not exist.

Properties Of Matrix Multiplication

(i) Matrix multiplication is not commutative in genral.

AB ≠ BA

(ii) Matrix multiplication is associative i.e.

(AB)C = A(BC), Whenever both sides are defined

(iii) Matrix multiplication is distributive over matrix addition

i.e. (a) A(B + C) = AB + AC

(b) (A + B)C = AC + AC

Whenever both sides of equality are defined.

(iv) If A is an m×n matrix, then

$I_mA = A = AI_n$

(v) The product of two matrices can be the null matrix while neither of them is the null matrix.

Example:- Let $A = \begin{bmatrix} 0 & 2 \\0 & 0 \end{bmatrix} \text{and} B = \begin{bmatrix} 1& 0 \\0 & 0\end{bmatrix}$

Then, $AB = \begin{bmatrix} 0 & 0 \\ 0 & 0\end{bmatrix}$ while neither A nor B is the null matrix.

(vi) If A is m×n matrix and o is a null matrix, then

(a) $A_{m\time n}O_{n\times p} = O_{m \times p}$

(b) $O_{p\times m}A_{m\times n} = O_{p\times n}$

(vii) In case of matrix multiplication if AB = O, then it does not necessarily imply that BA = O.

Example:- Let $A = \begin{bmatrix} 0 & 1 \\0 & 0 \end{bmatrix} \text{and} \begin{bmatrix} 1 & 0 \\ 0 & 0\end{bmatrix}$ . Then AB = O. But

$BA = \begin{bmatrix} 0 & 1 \\0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0\end{bmatrix}$

$= \begin{bmatrix} 0& 1 \\ 0 & 0\end{bmatrix}$ ≠ O

Thus, AB = O while BA ≠ O.

Positve Integral Power of a Square Matrix

Let A be a square matrix. Then we define

(i) A¹ = A

(ii) $A^{n+1} = A^n A$

(iii) $(A^m)^n = A^{mn}$ for all m, n ∈ N

Matrix Polynomial

Let $f(x) = a_0 x^n + a_1 x^{n-1} + a_2 x^{x - 2}+ . . . +a_{n-1} x + a_n$ be a polynomial and let A be a square matrix of order n. Then

$f(A) = a_0 A^n + a_1 A^{n-1} + a_2 A^{x - 2}+ . . . +a_{n-1} A + a_nI_n$

is called a polynomial matrix.

Question related to matrix

Question 1: Matrix Theory introduced by ………..

Answer: Matrix theory introduced by Cayley-Hamilton .

Question 2: Can a diagonal matrix is both an upper triangular and lower triangular matrix ?

Answer: Yes, a diagonal matrix is both an upper triangular and lower triangular matrix

Question 4: If S is a scalar matrix and A is a square matrix of the same order n, then AS = SA is true or false .

Answer : AS = SA is true for the statement.

Question 5: If A and B are two matrices such that AB and A + B are both defined, then A, B are square matrices of different order .

Answer: No, A and B matrices should be of same order

Some Other Solution

Class 12 ncert solution math exercise 3.1 matrix

Class 12 ncert solution math exercise 3.2 matrix

class 12 maths exercise 3.3 solution

class 12 maths chapter 3 Miscellaneous solution