This document provides a mathematics assignment on matrices and determinants containing 25 problems of varying difficulty levels. The problems involve calculating determinants, finding cofactors, inverses of matrices, solving systems of equations using matrices, expressing matrices as sums, and proving identities about determinants. Students are asked to perform operations on matrices, solve equations involving matrices, use properties of determinants, and apply concepts of invertibility and elementary row operations.

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
]