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

Question 3.

If F(α) = ⎡⎣⎢cosα0−sinα010sinα0cosα⎤⎦⎥ show that [F(α)]−1=F(−α)

Solution:

Let A = F (α)

So [F(α)]−1=A−1

Now

Question 4.

If A = [5−13−2] show that A^{2} – 3A – 7I_{2} = O_{2}. Hence find A^{-1}.

Solution:

A = [5−13−2]

To Find A-1

Now we have proved that A^{2} – 3A – 7I_{2} = O_{2}

Post multiply by A^{-1} we get

A – 3I – 7A^{-1} = O_{2}

Question 5.

If A=19⎡⎣⎢−84114−8474⎤⎦⎥ prove that A^{-1} = A^{T}

Solution:

Question 6.

If A=[8−5−43], verify that A(adj A) = (adj A)A = |A| I_{2}

Solution:

Question 7.

If A=[3725], and B=[−15−32] verify that (AB)^{-1} = B^{-1} A^{-1}.

Solution:

Question 8.

If adj (A) = ⎡⎣⎢2−3−2−41202−72⎤⎦⎥ find A

Solution:

Question 9.

If adj(A) = ⎡⎣⎢06−3−2200−66⎤⎦⎥ find A^{-1}

Solution:

Question 10.

Find adj(adj(A)) if adj A = ⎡⎣⎢10−1020101⎤⎦⎥

Solution:

Question 11.

Solution:

Question 12.

Find the matrix A for which A [5−13−2]=[14777]

Solution:

Given A [5−13−2]=[14777]

Let B=(5−13−2) 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=[12−10],B=[31−21] 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^{-1}CB^{-1} (i.e) X = A^{-1}CB^{-1}

Question 14.

Solution:

Question 15.

Decrypt the received encoded message [2−3][204] with the encryption matrix [−12−11] 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 [−12−11]

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

Thus the receivers read this message as HELP.

### 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: