Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.1

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.1

Question 1.
Find the adjoint of the following:

Solution:

Question 2.
Find the inverse (if it exists) of the following:

Solution:
For a matrix A, A1=1|A|(adjA). Where |A| ≠ 0. If |A| = 0 then A is called a singular matrix and so A1 does not exist.

Question 3.
If F(α) = ⎡⎣⎢cosα0sinα010sinα0cosα⎤⎦⎥ show that [F(α)]1=F(α)
Solution:
Let A = F (α)
So [F(α)]1=A1
Now

Question 4.
If A = [5132] show that A2 – 3A – 7I2 = O2. Hence find A-1.
Solution:
A = [5132]

To Find A-1
Now we have proved that A2 – 3A – 7I2 = O2
Post multiply by A-1 we get
A – 3I – 7A-1 = O2

Question 5.
If A=19⎡⎣⎢841148474⎤⎦⎥ prove that A-1 = AT
Solution:

Question 6.
If A=[8543], verify that A(adj A) = (adj A)A = |A| I2
Solution:

Question 7.
If A=[3725], and B=[1532] verify that (AB)-1 = B-1 A-1.
Solution:

Question 8.
If adj (A) = ⎡⎣⎢2324120272⎤⎦⎥ find A
Solution:

Question 9.
If adj(A) = ⎡⎣⎢063220066⎤⎦⎥ find A-1
Solution:

Question 10.
Solution:

Question 11.

Solution:

Question 12.
Find the matrix A for which A [5132]=[14777]
Solution:
Given A [5132]=[14777]
Let B=(5132) and C=(14777)
Given AB = C, To find A
Now AB = C
Post multiply by B-1 on both sides
ABB-1 = CB-1 (i.e) A (BB-1) = CB-1
⇒ A(I) = CB-1 (i.e) A = CB-1
To find B-1:

Question 13.
Given A=[1210],B=[3121] and C[1212], find a matrix X such that AXB = C.
Solution:
A × B = C
Pre multiply by A-1 and post multiply by B-1 we get
A-1 A × BB-1 = A-1CB-1 (i.e) X = A-1CB-1

Question 14.

Solution:

Question 15.
Decrypt the received encoded message [23][204] with the encryption matrix [1211] and the decryption matrix as its inverse, where the system of codes are described by the numbers 1-26 to the letters A- Z respectively, and the number 0 to a blank space.
Solution:
Let the encoding matrix be [1211]

So the sequence of decoded matrices is [8 5], [12 16].

Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.1 Additional Problems

Question 1.
Using elementary transformations find the inverse of the following matrix
Solution:

Question 2.
Using elementary transformations find the inverse of the matrix
Solution:

Question 3.
Using elementary transformation find the inverse of the matrix
Solution:

Question 4.
Using elementary transformations find the inverse of the matrix
Solution:

Question 5.
Using elementary transformation, find the inverse of the following matrix
Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution: