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NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.
NCERT solutions for Class 12 Maths Matrices Download as PDF
NCERT Solutions class 12 Maths Matrices
1. Let A = show that where I is the identity matrix of order 2 and N.
Ans. Using Mathematical Induction, we see the result is true for for
Given: is true, i.e.
To prove:
Proof: L.H.S. = = =
=
=
= = R.H.S.
Thus, is true, therefore, is true.
2. If A = , prove that A’’ = N.
Ans. Given: A = ……….(i)
Let
=
is true for
Now ……….(ii)
Multiplying eq. (ii) by eq. (i),
=
Therefore, is true for all natural numbers by P.M.I.
3. If A = then prove that A’’ = where is any positive integer.
Ans. Given: A’’ =
which is true for
Now, ……….(i)
Again ……….(ii)
[From eq. (i)]
=
Therefore, the result is true for
Hence, by the principal of mathematical induction, the result is true for all positive integers
4. If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
Ans. A and B are symmetric matrices. A’ = A and B’ = B ……….(i)
Now, (AB – BA)’ = (AB)’ – (BA)’ (AB – BA)’ = B’A’ – A’B’ [Reversal law]
(AB – BA)’ = BA – AB [Using eq. (i)]
(AB – BA)’ = – (AB – BA)
Therefore, (AB – BA) is a skew symmetric.
NCERT Solutions class 12 Maths Miscellaneous
5. Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Ans. (B’AB)’ = [B’(AB]’ = (AB)’ (B’)’ [ (CD)’ = D’C’]
(B’AB)’ = B’A’B ……….(i)
Case I: A is a symmetric matrix, then A’ = A
From eq. (i) (B’AB)’ = B’AB
B’AB is a symmetric matrix.
Case II: A is a skew symmetric matrix. A’ = – A
Putting A’ = – A in eq. (i), (B’AB)’ = B’(– A)B = – B’AB
B’AB is a skew symmetric matrix.
6. Find the values of if the matrix A = satisfies the equation A’A = I.
Ans. Given: A =
A’ =
Now A’A = I
Equating corresponding entries, we have
And
And
, ,
7. For what value of ?
Ans. Given:
Equating corresponding entries, we have
8. If A = show that A^{2} – 5A + 7I = 0.
Ans. Given: A =
A^{2} – 5A + 7I =
= =
= = =
= = 0 = R.H.S. Proved.
9. Find if
Ans. Given:
Equating corresponding entries, we have
10. A manufacturer produces three products, which he sells in two markets. Annual sales are indicated below:
| Market | Products |
I. 10,000 | 2,000 | 18,000 |
II. 6,000 | 20,000 | 8,000 |
(a) If unit sales prices of and are ` 2.50, ` 1.50 and ` 1.00 respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.
Ans. According to question, the matrix A =
(a) Let B be the column matrix representing sale price of each unit of products
Then B =
Now Revenue = Sale price c Number of items sold
= =
Therefore, the revenue collected by sale of all items in Market I = ` 46,000 and the revenue collected by sale of all items in Market II = ` 53,000.
(b) Let C be the column matrix representing cost price of each unit of products
Then C =
Total cost = AC =
= =
The profit collected in two markets is given in matrix form as
Profit matrix = Revenue matrix – Cost matrix
Therefore, the gross profit in both the markets = ` 15000 + ` 17000 = ` 32,000.
NCERT Solutions class 12 Maths Miscellaneous
11. Find the matrix X so that X
Ans. Given: X ……….(i)
Putting X = in eq. (i),
Equating corresponding entries, we have
…..(ii) …..(iii)
…..(iv) …..(v)
…..(vi) …..(vi)
Solving eq. (ii) and (iii), we have and
Solving eq. (v) and (vi), we have and
Putting these values in X = , X =
12. If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB’’ = B’’A. Further prove that (AB)’’ = A’’B’’ for all N.
Ans. Given: AB = BA …..(i)
Let ……(ii)
For becomes AB = BA
is true for
For
Multiplying both sides by B,
[From eq. (i)]
is also true.
Therefore, is true for all N by P.M.I.
NCERT Solutions class 12 Maths Miscellaneous
13. If A = is such that A^{2} = I, then:
(A)
(B)
(C)
(D)
Ans. Given: A = and A^{2} = I
Equating corresponding entries, we have
Therefore, option (C) is correct.
14. If the matrix A is both symmetric and skew symmetric, then:
(A) A is a diagonal matrix
(B) A is a zero matrix
(C) A is a square matrix
(D) None of these
Ans. Since, A is symmetric, therefore, A’ = A ……..(i)
And A is skew-symmetric, therefore, A’ = – A
A = – A [From eq. (i)]
A + A = 0 2A = 0 A = 0
Therefore, A is zero matrix.
Therefore, option (B) is correct.
15. If A is a square matrix such that A^{2} = A, then (I + A)^{3} – 7A is equal to:
(A) A
(B) I – A
(C) I
(D) 3A
Ans. Given: A^{2} = A …..(i)
Multiplying both sides by A, A^{3} = A^{2} = A [From eq. (i)] ……(ii)
Also given (I + A)^{3} – 7A = I^{3} + A^{3} + 3I^{2}A + 3IA^{2} – 7A
Putting A^{2} = A [from eq. (i)] and A^{3} = A [from eq. (ii)],
= I + A + 3IA + 3IA – 7A = I + A + 3A + 3A – 7A [ IA = A]
= I + 7A – 7A = I
Therefore, option (C) is correct.
NCERT Solutions class 12 Maths Miscellaneous
NCERT Solutions class 12 Maths Miscellaneous PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide
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