![CLASS XII MATHEMATICS ASSIGNMENT
MATRICES AND DETERMINANT
1 Mark Questions
1. A matrix A of order 3 x 3 has determinant 5. Find the value of |3𝐴| .
2. Find the cofactor of a12 in the following : |
1 −3 5
6 0 4
1 5 −7
| .
3. If A is a square matrix of order 3 such that | 𝑎𝑑𝑗 𝐴| = 64, 𝑓𝑖𝑛𝑑 | 𝐴| .
4. If A is a square matrix and satisfies the relation A2
+ A – I = 0 , then find A-1
.
5. For what value of a, [
2𝑎 −1
−8 3
] is singular matrix?
6. A square matrix ‘ A’ , of order 3 has | 𝐴| = 5, find | 𝐴 𝑎𝑑𝑗 𝐴| .
7. For what value of k, the matrix [
2 − 𝑘 3
−5 1
] is not invertible.
8. A matrix A is of order 2 x 2 has determinant 4. What is the value of|2𝐴| ?
9. If A and B are symmetric matrices of same order, write whether AB – BA is symmetric or skew symmetric.
4 Marks Questions
10. If A = [
2 3
1 −4
] and B = [
1 −2
−1 3
] and B = [
1 −2
−1 3
] , show (AB)-1
= B-1
A-1
.
11. If A = [
3 2
1 1
] , find the value of a and b : A2
+ Aa + bI = 0. Hence find A-1
.
12. Express the matrix [
2 3 4
5 7 9
−2 1 1
] as the sum of symmetric and a skew symmetric matrix.
13. Find the values of x, y, z if A = [
0 2𝑦 𝑧
𝑥 𝑦 −𝑧
𝑥 −𝑦 𝑧
] satisfies the eqn A’A = I .
14. If A = [
3 −4
1 1
] . Prove An
= . [
1 + 2𝑛 −4𝑛
𝑛 1 − 2𝑛
] for all n∈ 𝑁.
15. If A = [
3 −5
−4 2
] , show A2
– 5A – 14I = 0 and hence find A-1
.
16. Find 2 x 2 matrix B : B [
1 −2
1 4
] = [
6 0
0 6
] .
17. .Using properties of determinants, prove : |
3𝑎 −𝑎 + 𝑏 −𝑎 + 𝑐
𝑎 − 𝑏 3𝑏 𝑐 − 𝑏
𝑎 − 𝑐 𝑏 − 𝑐 3𝑐
| = 3(a + b + c) (ab + bc + ca)
18. .Using properties of determinants, prove : |
𝑦 + 𝑧 𝑧 𝑦
𝑧 𝑧 + 𝑥 𝑥
𝑦 𝑥 𝑥 + 𝑦
| = 4xyz
19. Using properties of determinants,prove |
𝑎2 + 1 𝑎𝑏 𝑎𝑐
𝑏𝑎 𝑏2 + 1 𝑏𝑐
𝑐𝑎 𝑐𝑏 𝑐2 + 1
| = a2
+ b2
+ c2
+ 1.](https://image.slidesharecdn.com/matricesanddeterminantsassignment-180625064430/85/Matrices-and-determinants-assignment-1-320.jpg)
![20. Using properties of determinants, prove |
1 + 𝑎2 − 𝑏2 2𝑎𝑏 −2𝑏
2𝑎𝑏 1 − 𝑎2 + 𝑏2 2𝑎
2𝑏 −2𝑎 1 − 𝑎2 − 𝑏2
| = (1+a2
+b2
)3
.
21. If x,y,z are different and |
𝑥 𝑥2 1 + 𝑥3
𝑦 𝑦2 1 + 𝑦3
𝑧 𝑧2 1 + 𝑧3
| = 0, then show that 1 + xyz = 0.
6 Marks Questions:
22. Using matrices, solve the system of equations x + y + z = 3, x- 2y + 3z = 6, -3y + 2z = 0
23. Find A-1
, where A = [
1 −2 0
2 1 3
0 −2 1
]. Hence solve the equations x – 2y = 10, 2x + y + 3z = 8, -2y + z = 7.
24. If A = [
1 −1 0
2 3 4
0 1 2
] and B = [
2 2 −4
−4 2 −4
2 −1 5
] , find AB. Hence ,solve the system x – y = 3,
2x + 3y + 4z = 17, y + 2z = 7.
25. Using elementary transformations, find the inverse of the matrix(
1 3 −2
−3 0 −1
2 1 0
) .
26. Find the inverse of the following matrix using elementary operations.
A = [
1 2 −2
−1 3 0
0 −2 1
]](https://image.slidesharecdn.com/matricesanddeterminantsassignment-180625064430/85/Matrices-and-determinants-assignment-2-320.jpg)
Matrices and determinants assignment
- 1. CLASS XII MATHEMATICS ASSIGNMENT MATRICES AND DETERMINANT 1 Mark Questions 1. A matrix A of order 3 x 3 has determinant 5. Find the value of |3𝐴| . 2. Find the cofactor of a12 in the following : | 1 −3 5 6 0 4 1 5 −7 | . 3. If A is a square matrix of order 3 such that | 𝑎𝑑𝑗 𝐴| = 64, 𝑓𝑖𝑛𝑑 | 𝐴| . 4. If A is a square matrix and satisfies the relation A2 + A – I = 0 , then find A-1 . 5. For what value of a, [ 2𝑎 −1 −8 3 ] is singular matrix? 6. A square matrix ‘ A’ , of order 3 has | 𝐴| = 5, find | 𝐴 𝑎𝑑𝑗 𝐴| . 7. For what value of k, the matrix [ 2 − 𝑘 3 −5 1 ] is not invertible. 8. A matrix A is of order 2 x 2 has determinant 4. What is the value of|2𝐴| ? 9. If A and B are symmetric matrices of same order, write whether AB – BA is symmetric or skew symmetric. 4 Marks Questions 10. If A = [ 2 3 1 −4 ] and B = [ 1 −2 −1 3 ] and B = [ 1 −2 −1 3 ] , show (AB)-1 = B-1 A-1 . 11. If A = [ 3 2 1 1 ] , find the value of a and b : A2 + Aa + bI = 0. Hence find A-1 . 12. Express the matrix [ 2 3 4 5 7 9 −2 1 1 ] as the sum of symmetric and a skew symmetric matrix. 13. Find the values of x, y, z if A = [ 0 2𝑦 𝑧 𝑥 𝑦 −𝑧 𝑥 −𝑦 𝑧 ] satisfies the eqn A’A = I . 14. If A = [ 3 −4 1 1 ] . Prove An = . [ 1 + 2𝑛 −4𝑛 𝑛 1 − 2𝑛 ] for all n∈ 𝑁. 15. If A = [ 3 −5 −4 2 ] , show A2 – 5A – 14I = 0 and hence find A-1 . 16. Find 2 x 2 matrix B : B [ 1 −2 1 4 ] = [ 6 0 0 6 ] . 17. .Using properties of determinants, prove : | 3𝑎 −𝑎 + 𝑏 −𝑎 + 𝑐 𝑎 − 𝑏 3𝑏 𝑐 − 𝑏 𝑎 − 𝑐 𝑏 − 𝑐 3𝑐 | = 3(a + b + c) (ab + bc + ca) 18. .Using properties of determinants, prove : | 𝑦 + 𝑧 𝑧 𝑦 𝑧 𝑧 + 𝑥 𝑥 𝑦 𝑥 𝑥 + 𝑦 | = 4xyz 19. Using properties of determinants,prove | 𝑎2 + 1 𝑎𝑏 𝑎𝑐 𝑏𝑎 𝑏2 + 1 𝑏𝑐 𝑐𝑎 𝑐𝑏 𝑐2 + 1 | = a2 + b2 + c2 + 1.
- 2. 20. Using properties of determinants, prove | 1 + 𝑎2 − 𝑏2 2𝑎𝑏 −2𝑏 2𝑎𝑏 1 − 𝑎2 + 𝑏2 2𝑎 2𝑏 −2𝑎 1 − 𝑎2 − 𝑏2 | = (1+a2 +b2 )3 . 21. If x,y,z are different and | 𝑥 𝑥2 1 + 𝑥3 𝑦 𝑦2 1 + 𝑦3 𝑧 𝑧2 1 + 𝑧3 | = 0, then show that 1 + xyz = 0. 6 Marks Questions: 22. Using matrices, solve the system of equations x + y + z = 3, x- 2y + 3z = 6, -3y + 2z = 0 23. Find A-1 , where A = [ 1 −2 0 2 1 3 0 −2 1 ]. Hence solve the equations x – 2y = 10, 2x + y + 3z = 8, -2y + z = 7. 24. If A = [ 1 −1 0 2 3 4 0 1 2 ] and B = [ 2 2 −4 −4 2 −4 2 −1 5 ] , find AB. Hence ,solve the system x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7. 25. Using elementary transformations, find the inverse of the matrix( 1 3 −2 −3 0 −1 2 1 0 ) . 26. Find the inverse of the following matrix using elementary operations. A = [ 1 2 −2 −1 3 0 0 −2 1 ]