Question 1.
Find the co-factors of the elements of the following matrices
Here, a11 = -11, M11 = 4
∴ A11 = (-1)1+1(4) = 4
a12 = 2, M12 = -3
∴ A12 = (-1)1+2(- 3) = 3
a21 = – 3, M21 = -2
∴ A21 = (- 1)2+1(2) = -2
a22 = 4, M22 = -1
∴ A22 = (-1)2+2(-1) = -1.
The co-factor of aij is given by Aij = (-1)i+jMij
Question 2.
Find the matrix of co-factors for the following matrices
Here, a11 = 1, M11 = -1
∴ A11 = (-1)1+1(-1) = -1
a12 = 3, M12 = 4
∴ A12 = (-1)1+2(4) = -4
a21 = 4, M21 = 3
∴ A21 = (-1)2+1(3) = -3
a22 = -1, M22 = 1
∴ A22 = (-1)2+1(1) = 1
A11 = -14, A12 = -10, A13 = -6,
A21 = 6, A22 = -5, A23 = -3,
A31 = -2, A32 = -7, A33 = 1.
∴ the co-factor matrix
Question 3.
Find the adjoint of the following matrices.
Here, a11 = 2, M11= 5
∴ A11 = (-1)1+1(5) = 5
a12 = -3, M12 = 3
∴ A12 = (-1)1+2(3) = -3
a21 = 3, M21 = -3
∴ A A21 = (-1)2+1(-3) = 3
a22 = 5, M22 = 2
∴ A22 = (-1)2+1 = 2
Solution:
A11 = -3, A12 = -12, A13 = 6,
A21 = -1, A22 = 3, A23 = 2,
A31 = -11, A32 = -9, A33 = 1
From (1), (2) and (3), we get,
A(adj A) = (adj A)A = |A|∙I.
Note: This relation is valid for any non-singular matrix A.
Question 5.
Find the inverse of the following matrices by the adjoint method
∴ A-1 exists.
First we have to find the co-factor matrix
= [Aij]2×2, where Aij = (-1)i+jMij
Now, A11 = (-1)1+1M11 = 2
A12 = (-1)1+2M12 = -(-3) = 3
A21 = (-1)2+1M21 = -5
A22 = (-1)2+2M22 = -1
Hence, the co-factor matrix
|A| = = 6 + 8 = 14 ≠ 0
∴ A-1 exist
First we have to find the co-factor matrix
= [Aij] 2×2 where Aij = (-1)i+jMij
Now, A11 = (-1)1+1M11 = 3
A12 = (-1)1+2M = -4
A21 = (-2)2+1M21 = (-2) = 2
A22 = (-1)2+2M22 = 2
Hence the co-factor matrix
= 1(10 – 0) – 0 + 0
= 1(10) – 0 + 0
= 10 ≠ 0
∴ A-1 exists.
First we have to find the co-factor matrix
Question 6.
Find the inverse of the following matrices
![Chapter 2 Matrices Ex 2.2 34 word image 18873 37](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18873-37.png)
![Chapter 2 Matrices Ex 2.2 35 Maharashtra Board 12th Maths Solutions Chapter 2 Matrics Ex 2.2 14](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/maharashtra-board-12th-maths-solutions-chapter-2-m-89.png)
![Chapter 2 Matrices Ex 2.2 36 word image 18873 39](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18873-39.png)
![Chapter 2 Matrices Ex 2.2 37 word image 18873 40](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18873-40.png)
![Chapter 2 Matrices Ex 2.2 38 word image 18873 41](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18873-41.jpeg)
![Chapter 2 Matrices Ex 2.2 39 Maharashtra Board 12th Maths Solutions Chapter 2 Matrics Ex 2.2 16](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/maharashtra-board-12th-maths-solutions-chapter-2-m-90.png)
![Chapter 2 Matrices Ex 2.2 40 Maharashtra Board 12th Maths Solutions Chapter 2 Matrics Ex 2.2 17](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/maharashtra-board-12th-maths-solutions-chapter-2-m-91.png)
![Chapter 2 Matrices Ex 2.2 41 word image 18873 44](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18873-44.png)
![Chapter 2 Matrices Ex 2.2 42 word image 18873 45](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18873-45.jpeg)
![Chapter 2 Matrices Ex 2.2 43 Maharashtra Board 12th Maths Solutions Chapter 2 Matrics Ex 2.2 19](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/maharashtra-board-12th-maths-solutions-chapter-2-m-92.png)
![Chapter 2 Matrices Ex 2.2 44 word image 18873 47](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18873-47.jpeg)