.
Chapter 3 STD 12 Date : 16/12/25
Matrices Maths
//X Section A
• Write the answer of the following questions. [Each carries 3 Marks] [45]
1. 3) Express the given matrix as the sum of a symmetric and a skew symmetric matrix :
é3 3 -1ù
ê-2 -2 1 ú
ê ú
êë-4 -5 2 úû
é 0 - tan a2 ù
2. If A = ê ú and I is the identity
êtan a 0 úû
ë 2
écos a - sin a ù
matrix of order 2, show that I + A = (I – A) ê sin a cos a ú .
ë û
é 1 0 2ù
ê ú
3. If A = ê0 2 1 ú then, prove that A3 – 6A2 + 7A + 2I = 0.
êë 2 0 3úû
4. 2) Express the given matrix as the sum of a symmetric and a skew symmetric matrix :
é 6 -2 2 ù
ê-2 3 -1ú
ê ú
êë 2 -1 3 úû
5. Show that the matrix B¢AB is symmetric or skew symmetric according as A is symmetric or skew
symmetric.
é3 3 2ù é 2 -1 2 ù
6. If A = ê ú and B = ê1 2 4 ú , verify that (i) (A')' = A (ii) (A + B)' = A' + B' (iii) (kB)' = kB',
ë4 2 0û ë û
where k is any constant.
é 3 -2 ù é1 0 ù 2
7. If A = ê ú and I = ê ú , find k so that A = kA – 2I.
ë 4 - 2 û ë 0 1 û
é1 2 3ù é-7 -8 -9ù
8. Find the matrix X so that, X ê ú = ê 4 6 úû
.
ë4 5 6û ë 2
9. A trust fund has ` 30,000 that must be invested in two different types of bonds. 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 among the two types of bonds. If the trust fund must obtain an
annual total interest of : (a) ` 1800 (b) ` 2000
é 2 3ù é 2 -2ù
7.2)
10. Find X and Y, if 2X + 3Y = ê ú and 3X + 2Y = ê ú
ë4 0 û ë -1 5 û
é0 a bù
1 1
11. Find (A + A') and (A – A'), when A = ê - a 0 c úú .
ê
2 2
êë -b - c 0 úû
é -2 ù
12. A = ê 4 ú,
If ê ú B = [1 3 –6], verify that (AB)' = B'A'.
êë 5 úû
écos x - sin x 0ù
13. If F( x ) = êê sin x cos x 0úú , show that F(x) F(y) = F(x + y).
êë 0 0 1úû
14. Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati,
Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month
of September and October are given by the following matrices A and B.
September Sales (in Rupees)
, Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati,
Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month
of September and October are given by the following matrices A and B.
September Sales (in Rupees)
Basmati Permal Naura
é10,000 20,000 30,000 ù Ramkishan
A = ê ú
ë50,000 30,000 10,000 û Gurcharan Singh
October Sales (in Rupees)
é 5,000 10,000 6,000 ù Ramkishan
Basmati Permal Naura B = ê 20,000 10,000 10,000ú Gurcharan Singh
ë û
(i) Find the combined sales in September and October for each farmer in each variety.
(ii) Find the decrease in sales from September to October.
(iii)If both farmers receive 2% profit on gross sales, compute the profit for each farmer and for each
variety sold in October.
é3 5ù
15. 1) Express the given matrix as the sum of a symmetric and a skew symmetric matrix : ê
ë1 -1úû
//X Section B
• Write the answer of the following questions. [Each carries 4 Marks] [52]
é1 1 1ù
ê ú
16. For the matrix A = ê1 2 -3ú , Show that A3 – 6A2 + 5A + 11 I = O. Hence, find A–1.
êë 2 -1 3 úû
é1 0 2 ù é x ù
17. Find x, if [x –5 –1] ê0 2 1 ú ê 4 ú = O.
ê ú ê ú
êë2 0 3úû êë 1 úû
é1 0 2 ù
18. For matrix A = êê0 2 1 úú show that A3 – 6A2 + 7A + 2I = O and hence find A–1.
êë 2 0 3úû
19. A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are
indicated below :
Products
Market
x y z
I 10,000 2,000 18,000
II 6,000 20,000 8,000
(a) If unit sale prices of x, y and z 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.
é0 2y z ù
20. Find the values of x, y, z if the matrix A = êê x y - z úú satisfy the equation A'A = I.
ëê x -y z úû
é cos q sin q ù é cos nq sin nq ù
If A = ê An = ê ú , n Î N.
21. ú , then prove that
ë - sin q cos q û ë - sin nq cos nqû
é 2 -1ù é5 2ù é2 5ù
22. Let A = ê ú , B=ê ú , C =ê ú . Find a matrix D such that CD – AB = O..
ë3 4 û ë7 4û ë3 8û
23. If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
24. 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.
, é 2 -2 -4 ù
25. Express the matrix B = ê -1 3 4 ú as the sum of a symmetric and a skew symmetric matrix.
ê ú
êë 1 -2 -3 úû
é2 0 1ù
26. Find A2 – 5A + 6I, if A = êê 2 1 3úú .
êë1 -1 0 úû
é0 6 7ù é0 1 1 ù é2ù
ê ú ê ú ê ú
27. If A = ê -6 0 8 ú , B = ê1 0 2ú , C = ê -2 ú. Calculate AC, BC and (A + B)C. Also, verify that (A + B)C
ëê 7 -8 0 ûú ëê 1 2 0 ûú ëê 3 ûú
= AC + BC.
é1 1 -1ù é 1 3ù
= ê 2 0 3 ú , B = ê 0 2ú é 1 2 3 -4 ù
A ú and C = ê 2 0 -2 1 ú , find A(BC), (AB)C and show that (AB)C
28. If ê ú ê
êë 3 -1 2 úû êë -1 4úû ë û
= A(BC).
Chapter 3 STD 12 Date : 16/12/25
Matrices Maths
//X Section A
• Write the answer of the following questions. [Each carries 3 Marks] [45]
1. 3) Express the given matrix as the sum of a symmetric and a skew symmetric matrix :
é3 3 -1ù
ê-2 -2 1 ú
ê ú
êë-4 -5 2 úû
é 0 - tan a2 ù
2. If A = ê ú and I is the identity
êtan a 0 úû
ë 2
écos a - sin a ù
matrix of order 2, show that I + A = (I – A) ê sin a cos a ú .
ë û
é 1 0 2ù
ê ú
3. If A = ê0 2 1 ú then, prove that A3 – 6A2 + 7A + 2I = 0.
êë 2 0 3úû
4. 2) Express the given matrix as the sum of a symmetric and a skew symmetric matrix :
é 6 -2 2 ù
ê-2 3 -1ú
ê ú
êë 2 -1 3 úû
5. Show that the matrix B¢AB is symmetric or skew symmetric according as A is symmetric or skew
symmetric.
é3 3 2ù é 2 -1 2 ù
6. If A = ê ú and B = ê1 2 4 ú , verify that (i) (A')' = A (ii) (A + B)' = A' + B' (iii) (kB)' = kB',
ë4 2 0û ë û
where k is any constant.
é 3 -2 ù é1 0 ù 2
7. If A = ê ú and I = ê ú , find k so that A = kA – 2I.
ë 4 - 2 û ë 0 1 û
é1 2 3ù é-7 -8 -9ù
8. Find the matrix X so that, X ê ú = ê 4 6 úû
.
ë4 5 6û ë 2
9. A trust fund has ` 30,000 that must be invested in two different types of bonds. 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 among the two types of bonds. If the trust fund must obtain an
annual total interest of : (a) ` 1800 (b) ` 2000
é 2 3ù é 2 -2ù
7.2)
10. Find X and Y, if 2X + 3Y = ê ú and 3X + 2Y = ê ú
ë4 0 û ë -1 5 û
é0 a bù
1 1
11. Find (A + A') and (A – A'), when A = ê - a 0 c úú .
ê
2 2
êë -b - c 0 úû
é -2 ù
12. A = ê 4 ú,
If ê ú B = [1 3 –6], verify that (AB)' = B'A'.
êë 5 úû
écos x - sin x 0ù
13. If F( x ) = êê sin x cos x 0úú , show that F(x) F(y) = F(x + y).
êë 0 0 1úû
14. Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati,
Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month
of September and October are given by the following matrices A and B.
September Sales (in Rupees)
, Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati,
Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month
of September and October are given by the following matrices A and B.
September Sales (in Rupees)
Basmati Permal Naura
é10,000 20,000 30,000 ù Ramkishan
A = ê ú
ë50,000 30,000 10,000 û Gurcharan Singh
October Sales (in Rupees)
é 5,000 10,000 6,000 ù Ramkishan
Basmati Permal Naura B = ê 20,000 10,000 10,000ú Gurcharan Singh
ë û
(i) Find the combined sales in September and October for each farmer in each variety.
(ii) Find the decrease in sales from September to October.
(iii)If both farmers receive 2% profit on gross sales, compute the profit for each farmer and for each
variety sold in October.
é3 5ù
15. 1) Express the given matrix as the sum of a symmetric and a skew symmetric matrix : ê
ë1 -1úû
//X Section B
• Write the answer of the following questions. [Each carries 4 Marks] [52]
é1 1 1ù
ê ú
16. For the matrix A = ê1 2 -3ú , Show that A3 – 6A2 + 5A + 11 I = O. Hence, find A–1.
êë 2 -1 3 úû
é1 0 2 ù é x ù
17. Find x, if [x –5 –1] ê0 2 1 ú ê 4 ú = O.
ê ú ê ú
êë2 0 3úû êë 1 úû
é1 0 2 ù
18. For matrix A = êê0 2 1 úú show that A3 – 6A2 + 7A + 2I = O and hence find A–1.
êë 2 0 3úû
19. A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are
indicated below :
Products
Market
x y z
I 10,000 2,000 18,000
II 6,000 20,000 8,000
(a) If unit sale prices of x, y and z 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.
é0 2y z ù
20. Find the values of x, y, z if the matrix A = êê x y - z úú satisfy the equation A'A = I.
ëê x -y z úû
é cos q sin q ù é cos nq sin nq ù
If A = ê An = ê ú , n Î N.
21. ú , then prove that
ë - sin q cos q û ë - sin nq cos nqû
é 2 -1ù é5 2ù é2 5ù
22. Let A = ê ú , B=ê ú , C =ê ú . Find a matrix D such that CD – AB = O..
ë3 4 û ë7 4û ë3 8û
23. If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
24. 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.
, é 2 -2 -4 ù
25. Express the matrix B = ê -1 3 4 ú as the sum of a symmetric and a skew symmetric matrix.
ê ú
êë 1 -2 -3 úû
é2 0 1ù
26. Find A2 – 5A + 6I, if A = êê 2 1 3úú .
êë1 -1 0 úû
é0 6 7ù é0 1 1 ù é2ù
ê ú ê ú ê ú
27. If A = ê -6 0 8 ú , B = ê1 0 2ú , C = ê -2 ú. Calculate AC, BC and (A + B)C. Also, verify that (A + B)C
ëê 7 -8 0 ûú ëê 1 2 0 ûú ëê 3 ûú
= AC + BC.
é1 1 -1ù é 1 3ù
= ê 2 0 3 ú , B = ê 0 2ú é 1 2 3 -4 ù
A ú and C = ê 2 0 -2 1 ú , find A(BC), (AB)C and show that (AB)C
28. If ê ú ê
êë 3 -1 2 úû êë -1 4úû ë û
= A(BC).
