Class 10 – Matrices

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Introduction: “MATRIX’ is a Latin word for womb. The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. Matrix notation and operations are used in electronic spread sheet programs in PC which can be used in different areas of business and science. Matrices are also used in cryptography.

Matrix:A rectangular arrangement of set of elements in the form of horizontal and vertical lines is called matrix. The elements can be numbers or variables.

Example:

Numbers arranged in the form of horizontal lines are called rows. Numbers arranged in the form of vertical lines are called columns. Elements of a matrix are represented in pair of brackets (or) parenthesis.
i.e., [    ] (or) ( ). Matrices is the plural of matrix.

Generally, we use capital letters to denote matrices.

Example:

Matrix A contains 2 rows and 3 columns.

Order of a matrix:A matrix having ‘m’ rows and ‘n’ columns is said to be of order m × n (m cross n (or) m by n).

If matrix A has ‘m’ rows and ‘n’ columns then m × n (m cross n (or) m by n) is the order of matrix, and is denoted by Am×n.

Example:

Number of rows = 2

Number of columns = 2

Order of matrix = 2 × 2.

It is denoted by A2 × 2

Example:The order of the matrix  is:

Solution:Number of rows = 2.

Number of columns = 3.

Order of matrix B = 2 × 3.

It is denote by B2 × 3

Example: The order of matrix is:

Solution:Number of rows = 1.

Number of columns = 4.

Order of matrix C = 1 × 4.

It is denote by C1 × 4

Example: The order of matrix D = [ 1 ] is

Solution:Number of rows = 1 = Number of columns.

Order of matrix D = 1 × 1.

It is denote by D1 × 1

A matrix having m rows and n columns having mn (or) nm number of elements.

Example:

Order of matrix = 2 × 3

The element 6 occurs in first row and third column. Therefore it can be represented as a13 = 6.

General form of a matrix:

In general, m × n matrix can be represented as

 where

aij is the element of the matrix in ith row and jth column i.e., is called (i, j)th element of the matrix.

Example: Construct a 2 × 3 matrix whose elements are defined by  for the representation of elements.

Solution: Given

i = 1, 2; j = 1, 2, 3

Required matrix =

Example:Construct a 3 × 2 matrix whose elements are defined by

Solution:

i = 1, 2, 3; j = 1, 2

Required matrix

Example: Construct a 2 × 2 matrix if  for the representation of elements.

Solution: Given

i = 1, 2; j = 1, 2

Required matrix =

Types of matrices:

i)               Row matrix: A matrix having only one row is called row matrix.
Example:

Order of any row matrix is , where n is number of column, n = 2, 3, 4….

ii)             Column matrix:A matrix having only one column is called column matrix.
Example:

Order of any column matrix is , where m is number of column, m = 2, 3, 4….

iii)            Rectangular matrix: A matrix in which number of rows are not equal to number of columns is called rectangular matrix.
Example:

iv)            Square matrix: A matrix having equal number of rows and columns is called a square matrix.

is a square matrix if m = n. A matrix of order

is called a square matrix of order m.
Example:


If is a square matrix of order n, then the elements a11, a22, …. ann constitute principal diagonal. Hence aij is an element of the diagonal if i = j (or) non-diagonal if i ≠ j.

Example: 

, a, e, k are the elements of the principal diagonal.

The sum of elements of the diagonal of a square matrix. A is called trace of A is denoted by Tr(A).
Example:
, find the trace of A?
Solution: Elements of principal diagonal are 1, 0, 9.
Tr(A) = 1 + 0 + 9 = 10.

v)             Diagonal matrix: If each non-diagonal element of a square matrix is equal to zero, then the matrix is called a diagonal matrix.
If
is a diagonal matrix, then . It is denoted as .
Example:

Example:Determine the diagonal elements of the matrix

Solution:i = j

a11 = a, a22 = B, a33 = C1

Diagonal elements of the matrix are a, B, C1.

Example: Determine the principal diagonal of the matrix given by

 of order 3 × 3

Solution: Given

vi)            Scalar matrix: If each non-diagonal element of a square matrix is zero and all diagonal elements are same, then it is called a scalar matrix.
A matrix is said to be a scalar matrix if.

Example:

vii)          Unit matrix (or) Identity matrix:If each non-diagonal element of a square matrix is zero and each diagonal element is equal to 1, then that matrix is called a unit or identity matrix. It is denoted by In (or) I.
If


Example:

Example:

viii)         Zero (or) Null matrix: If each element of a matrix is zero, then it is called a null matrix or zero matrix. It is denoted by Om×n (or) O.

Example:

A square matrix  is said to be upper triangular if aij = 0 for all i > j.
A square matrix
 is said to be lower triangular if aij = 0 for all i < j.

Example:

i)   upper triangular matrices.

 ii)  lower triangular matrices.

Comparison of matrices: Two matrices can be compared if the order of matrices is equal. i.e., they have same number of rows and same number of columns.

Example:

Compared because order of matrices is equal.

Equality of matrices: Matrices A and B are said to be equal if A and B are of same order and the corresponding elements of A and B are equal.

Thus, A = B if aij = bij

Example:

Example:If  such that A = B then find x, y, z.

Solution:

Given A = B, corresponding elements are equal.

Multiplication of a matrix by a scalar:
If every element of a matrix A is multiplied by a number (real or complex) k, the matrix obtained is k times A and is denoted by kA and the operation is called scalar multiplication.

Example:

1.             

Solution:

 

1.              If a and b are any two scalars and P is a matrix, then a(bP) = (ab)P

2.              If m and n are any two scalars and A is a matrix, then (m + n)A = mA + nA. 

Addition of matrices: Two matrices can be added if the order of matrices are equal.

The sum matrix of two matrices A and B is obtained by adding the corresponding elements of A and B.

If A =  and B =  then

A + B can be represented as  such that i = 1, 2, 3, ….. m, j =1, 2, 3, ….. n.

Example:

Example: Find the sum matrix of matrices  and

Solution:The given two matrices are of same order. So they can be added.

Sum =

Example: Find the sum of 2 × 2 matrices .

Solution:A = [aij]

i = 1, 2; j = 1, 2

B = [bij]

i = 1, 2; j = 1, 2

Properties on addition of matrices:

i)               Closure property: When we add any two matrices of same order then the resultant also is matrix of same order.
Example:

ii)             Commutative property:Let A, B be two matrices then A + B = B + A
Example:

iii)            Associative property: Let A, B, C be three matrices then (A + B) + C = A + (B + C)
Example:

iv)            Additive identity: Let A be a matrix and O is zero matrix then A + O = O + A = A
Example:

v)             Additive inverse: Let A be a matrix then A + (-A) = (-A) + A = O
(-A) is called additive inverse of A.
A is called additive inverse of –A.

Example:

vi)            Cancellation laws:
A + B = A + C
B = C (Left cancellation law)
B + A = C + A
B = C (Right cancellation law)

Subtraction of matrices: Two matrices can be subtracted if they are of the same order.

The difference of two matrices of same order A and B i.e. A – B is obtained by subtracting the corresponding elements of B from that of A. Also the difference matrix is of the same order as that of A or B.

If  then A – B can be represented as such that

i = 1, 2, 3, ……m; j =1, 2, 3, ……n.

Example:

Example: Find the subtraction of matrices

Solution:

i)

Matrix subtraction is not commutative.

ii)

 Matrix subtraction is not associative.

Example:Find the subtraction of matrices if

Solution: Given

Aliter:

Example: Find A + (-B) if

Solution:Given

A + (-B) = A – B

Solving a matrix equation: Let A, B be two matrices of same order such that A + X = B where X is an unknown matrix or order equal to A and B matrices.

Example: find X of order 2 × 2 such that

i) X = A – B                 ii) A + X = B

Solution: Given

i) X = A – B 

ii) A + X = B

Transpose of a matrix:The matrix obtained from any given matrix A by interchanging its rows and columns is called transpose of the given matrix. It is denoted by AT.
If
 then AT =

Example:
Solution:

Example:If  then find  matrix.

Solution:

Properties of transpose matrix:

i)               If A is any matrix then

ii)             If A, B are two matrices of same order then

iii)            If A, B are two matrices of same order then

iv)            If A is a matrix and k is a scalar then

Example:If  then find

i)                   ii)                   iii)

Solution: i) Given

ii)

iii)

Example: If  and k is a scalar then verify that

Solution:Given

Symmetric matrix: A square matrix is said to be symmetrix if the transpose of the matrix is equal to itself.

A square matrix is said to be symmetric if its (i, j)th element is same as its
(j, i)th element, i.e., aij = aji, for all i, j.

Example:

    A is symmetric matrix.

 

i)               Symmetric matrix is always a square matrix.

ii)             A necessary and sufficient condition for a matrix A to be symmetric is that it is equal to its transpose matrix. i.e., .

iii)            Diagonal matrices are always symmetric.

Skew-symmetric matrix: If a square matrix and the negative of its transpose matrix are equal then it is called a skew-symmetric matrix.

A square matrix A = [aij] is said to be skew-symmetric if the (i, j)th element of A is the negative of the (j, i)th element of A. i.e., aij = -aji, for all i, j.

Example:

A is skew symmetric matrix.

A matrix which is both symmetric and skew symmetric is called a square null matrix.