OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6. Step by step Solutions of OP Malhotra SK Gupta, Anubhuti Gangal S.Chand ISC Class-12 Mathematics with Exe-6(a), Exe-6(b), Exe-6(c), Exe-6(d), Exe-6(e), Exe-6(f), Exe-6(g), Exe-6(h), Exe-6(i), Exe-6(j), Self Revision and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Class:	12th
Subject:	Mathematics
Chapter  :	Ch-6 Matrices of Section -A

Board	ISC
Writer	OP Malhotra, SK Gupta, Anubhuti Gangal
Publications	S.Chand Publications 2020-21

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-: Included Topics :- 

Exe-6(a)

Exe-6(b)

Exe-6(c)

Exe-6(d)

Exe-6(e)

Exe-6(f)

Exe-6(g)

Exe-6(h)

Exe-6(i)

Exe-6(j)

Self Revision

Chapter Test

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OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Matrices Definition :

A matrix is a function which consists of an ordered rectangular array of numbers. The numbers in the array are called the entities or the elements of the matrix. The horizontal array of elements in the matrix is called rows, and the vertical array of elements are called the columns. If a matrix has m rows and n columns, then it is known as the matrix of order m x n.

or

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

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Exe-6(a)

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

If a matrix ha 8 element, what are the possible order it can have ? What if it has 5 elements ?

Question 2:

How many entries ………………………….. matrix of order n ?

Question 3:

……………………….

……………………….

……………………..

Question 12:

Find the value of a, b, c and d, if …………………..

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Exe-6(b)

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

Write each sum or different as a single matrix.

(i)……..

(ii)…….

(iii)…….

Question 2:

(i) Given A = ……………..

Question 3:

…………………

…………………

………………….

Question 15:

Let A = ……………………. find the matrix B ………..

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Types of Matrices :

Depending upon the order and elements, matrices are classified as:

Square Matrix :

A matrix which has an equal number of rows and columns. It is expressed as m × m.

Diagonal Matrix :

A matrix which has non-zero elements in its diagonal part running from upper left to the lower right or vice versa.

Scalar Matrix :

 The scalar matrix is a square matrix, which has all its diagonal elements equal and all the off-diagonal elements as zero.

Column Matrix :

A column matrix is a m × 1 matrix, consisting of a single column of m elements. It is also called a column vector.

Row Matrix :

A row matrix is a 1 × m matrix, consisting of a single row of m elements. It is also called a row vector.

Identity Matrix :

A square matrix which has all its principal diagonal elements as 1’s and all non-diagonal elements as zeros.

Zero Matrix :

A matrix whose all entries are zero. It is also called a null matrix.

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Exe-6(c)

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

(a) State whether each product is defined, If so, give its dimensions.

………………..

…………………

……………….

Question 2:

Calculate :

(i) ………………

……………………

…………………..

Question 3:

(a) find the value of x in the following :

………………..

………………….

…………………

Question 25:

If A square matrix such that A² = A ……………………………… is an identity matrix.

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Exe-6(d)

 Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

A man buy 8 doze mangoes, ………………………………………….  hence obtain the total cost.

Question 2:

A store has in stock ………………………………….. item in the stock.

Question 3:

……………………….

………………………….

…………………….

Question 6:

In a development plan of a city , …………………………………………  firm A, B, C respectively.

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Exe-6(e)

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

(i) If A = …………………..

(ii) For matrix A = …………………

(iii) Find 1/2 …………………..

Question 2:

If D = …………………………

Question 3:

………………………

…………………….

……………………..

Question 11:

If A = ………………………….. A’A – 1.

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Order of a Matrix :

If a matrix has m rows and n columns, then its order is written as m × n. If a matrix has order m × n, then it has mn elements.

In general, a_m×n matrix has the following rectangular array:

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Exe-6(f)

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

Define a symmetric matrix, show that he following matrices are symmeric.

………………

……………..

Question 2:

Show that each of the following is a skew matrix.

…………………

Question 3:

……………………………

……………………………

…………………………..

Question 9:

Express the following matrix as the sum of a symmetric ………………………….. verify your result.

……………………

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Properties of Addition of Matrices :
(a) Commutative If A = [a_ij] and B = [b_ij] are matrices of the same order say m x n then A + B = B + A,
(b) Associative for any three matrices A = [a_ij], B = [b_ij], C = [c_ij] of the same order say m x n, A + (B + C) = (A + B) + C.
(c) Existence of additive identity Let A = [aij] be amxn matrix and O be amxn zero matrix, then A + O = O + A = A. In other words, O is the additive identity for matrix addition.
(d) Existence of additive inverse Let A = [a_ij]_m×n be any matrix, then we have another matrix as -A = [-a_ij]_m×n such that A + (-A) = (-A + A) = O. So, matrix (-A) is called additive inverse of A or negative of A.

Note :
(i) If A and B are not of the same order, then A + B is not defined.
(ii) Addition of matrices is an example of a binary operation on the set of matrices of the same order.

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Exe-6(g)

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

Let A be a square matrix ………………………………… where |A| = 4

Question 2:

If A is square matrix of order 3 …………………………….. find |A|.

Question 3:

…………………….

……………………..

……………………

Question 14:

Prove that

|adj AB| = |adj A| |adj B|.

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Exe-6(h)

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

Show that A is a singular matrix If

(i)………………

(ii)………………

Question 2:

Find x if ………………………. is a singular matrix.

Question 22:

If A = 1/9 ………………….. prove that A…………………

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Multiplication of Matrices : 

Let A and B be two matrices. Then, their product AB is defined, if the number of columns in matrix A is equal to the number of rows in matrix B.

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Exe-6(i)

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Using elementary transformation, find the inverse, find the inverse of the following matrices, if it exits.

Question 1:

A = …………….

Question 2:

……………………

……………………….

…………………….

Question 7:

……………………….

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Exe-6(j)

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Solve the following systems of equations by matrix method.

Question 1:

2x – 3y = 1

3x – 2y = 4

Question 2:

……………………..

………………………

…………………….

Question 22:

A school wants to award its students …………………………………… using matrix method.

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Properties of Multiplication of Matrices :

(a) Associative law For three matrices A, B, and C, if multiplication is defined, then A (BC) = (AB) C.

(b) Non-commutativity Matrix multiplication is not commutative i.e. if AB and BA are both defined, then it is not necessary that AB ≠ BA.

(c) Multiplicative identity For every square matrix A, there exists an identity matrix of the same order such that IA = AI = A.

Note:

For Amxm, there is only one multiplicative identity I_m.
(d) Distributive law For three matrices A, B, and C,
A(B + C) = AB + AC
(A + B)C = AC + BC
whenever both sides of the equality are defined.

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Self Revision

 Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

Using matrices, solve the following system of equation:

……………………

…………………..

Question 2:

Using matrices, solve the following system of equation:

………………..

…………………

Question 3:

………………………..

………………………….

……………………….

Question 29:

solve the following system of linear equations using matrix method.

…………………….

…………………….

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Chapter Test

OP Malhotra Matrices S.Chand ISC Class-12 Maths Solutions Ch-6

Question 1:

(a) A = ……………………

(b) Find the value of x and y if ………………….

………………

………………..

Question 2:

…………………..

…………………..

……………………

Question 14:

Using elementary transformation, find the inverse …………………….. of linear equations

…………………

-: End of Matrices OP Malhotra S. Chand ISC Class-12 Maths Chapter-6 Solution :-

Return to :-  OP Malhotra S. Chand ISC Class-12 Maths Solutions

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