Exercise 3.2
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NCERT Solutions class 12 Maths Matrices
1. Let A = 
B = 
C = 
. Find each of the following:
(i) A + B
(ii) A – B
(iii) 3A – C
(iv) AB
(v) BA
Ans. (i) A + B = 
= 
(ii) A – B = 
= 
(iii) 3A – C = 
= 
(iv) AB = 
= 
(v) BA = 
= 
2. Compute the following:
(i) 
(ii) 
(iii) 
(iv) 
Ans. (i) = 
(ii)
= 
(iii) = 
(iv) = 
3. Compute the indicated products:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Ans. (i) = 
(ii) = 
(iii) = 
(iv)
= 
= 
(v)
= 
(vi)
= 
4. If A = 
B = 
and C = 
then compute (A + B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.
Ans. A + B = 
= 
= 
B – C = 
= 
= 
Now, A + (B – C) = (A + B) – C
= 
= 
=
L.H.S. = R.H.S. Proved.
5. If A = 
and B = 
then compute 3A – 5B.
Ans. 3A – 5B = 
= 
= 
6. Simplify: 
Ans. Given:
= 
= 
7. Find X and Y, if:
(i) X + Y = 
and X – Y = 
(ii) 2X + 3Y = 
and 3X + 2Y = 
Ans. (i) Given: X + Y = …..(i)
and X – Y = …..(ii)
Adding eq. (i) and (ii), we get
2X = 
X = 
Subtracting eq. (i) and (ii), we get
2Y = 
Y = 
(ii) Given: 2X + 3Y = …..(i)
and 3X + 2Y = 
…..(ii)
Multiplying eq. (i) by 2, 4X + 6Y = 
……….(iii)
Multiplying eq. (ii) by 3, 9X + 6Y = 
………(iv)
Eq. (iv) – Eq. (iii) = 5X = 
= 
X = 
Now, From eq. (i), 3Y = 
2X = 
3Y = 
= 
Y = 
8. Fin X if Y = 
and 2X + Y = 
Ans. 2X + Y = 
2X = – Y
2X = 
2X = 
X = 
= 
9. Find 
and 
if 
Ans. Given: 
Equating corresponding entries, we have
and 
and 
and 
and 
10. Solve the equation for 
and 
if 
Ans. Given: 
Equating corresponding entries, we have
And 
And 
And 
, , ,
11. If 
find the values of and 
Ans. Given: 
Equating corresponding entries, we have
……….(i) and 
……….(ii)
Adding eq. (i) and (ii), we have 
Putting in eq. (ii), 
12. Given: 
find the values of and 
Ans. Given: 
Equating corresponding entries, we have
And 
And 
……….(i)
And 
Putting in eq. (i), 
, , ,
13. If 
show that 
Ans. Given: 
……….(i)
Changing to 
in eq. (i), 
L.H.S. = 
= 
= 
= 
= R.H.S. [changing to 
in eq. (i)]
14. Show that:
(i) 
(ii) 
Ans. (i) L.H.S. = 
= 
= 
R.H.S. = 
= 
= 
L.H.S. 
R.H.S.
(ii) L.H.S. = 
= 
= 
R.H.S. = 
= 
= 
L.H.S. R.H.S.
15. Find A2 – 5A + 6I if A = 
.
Ans. A2 – 5A + 6I = 
= 
= 
= 
= 
16. If A = 
prove that A3 – 6A2 + 7A + 2I = 0.
Ans. L.H.S. = A3 – 6A2 + 7A + 2I
= 
= 
= 
= 
= 
= 
= 
= 
= 0 (Zero matrix)
= R.H.S. Proved.
17. If A = 
and I = 
find 
so that 
Ans. Given: A = and I =
Equating corresponding entries, we have
And 
and 
NCERT Solutions class 12 Maths Exercise 3.2
18. If A = 
and I is the identity matrix of order 2, show that
Ans. L.H.S. = I + A = 
Now, I – A = 
R.H.S. = 
= 
= 
= 
= 
= 
= 
= 
L.H.S. = R.H.S. Proved.
19. A trust fund has ` 30,000 that must be invested in two different types of bond. The first bond pays 5% interest per year and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide ` 30,000 in two types of bonds, if the trust fund must obtain an annual interest of (a) ` 1800, (b) ` 2000.
Ans. Let the investment in first bond be ` , then the investment in the second bond = `
Interest paid by first bond = 5% = 
per rupee and interest paid by second bond = 5% = 
per rupee.
Matrix of investment is A = 
Matrix of annual interest per rupee B = 
Matrix of total annual interest is AB = 
= 
= 
= 
Total annual interest = ` 
(a) According to question, 
Therefore, Investment in first bond = ` 15,000
And Investment in second bond = ` (30000 – 15000) = ` 15,000
(b) According to question, 
Therefore, Investment in first bond = ` 5,000
And Investment in second bond = ` (30000 – 15000) = ` 25,000
NCERT Solutions class 12 Maths Exercise 3.2
20. The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are ` 80, ` 60 and ` 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
Ans. Let the number of books as a 1 x 3 matrix B = 
Let the selling prices of each book as a 3 x 1 matrix S = 
Total amount received by selling all books = BS = 
= 
= 
Therefore, Total amount received by selling all the books = ` 20160
NCERT Solutions class 12 Maths Exercise 3.2
21. The restriction on 
and 
so that PY + WY will be define are:
(A) 
(B) is arbitrary, 
(C) is arbitrary, 
(D) 
Ans. Given: 
Now, 
On comparing, and 
Therefore, option (A) is correct.
NCERT Solutions class 12 Maths Exercise 3.2
22. If 
then order of matrix 7X – 5Z is:
(A) 
(B) 
(C) 
(D) 
Ans. Here 
(given), the order of matrices X and Z are equal.
7X – 5Z is well defined and the order of 7X – 5Z is same as the order of X and Z.
The order of 7X – 5Z is either equal to or 
But it is given that
Therefore, the option (B) is correct.