# EXERCISE 4.5 (Determinant)

**class 12 maths exercise 4.5 ncert solution**

**Question 1:** Find the adjoint of the matrix

**Solution:** Let

Then,

Thus

**Question 2:** Find the adjoint of the matrix

**Solution:** Let

Then,

**Question 3:** Verify for

**Solution:** Let

Then,

Also,

Now,

Hence,

Now,

Also,

Hence,

.

**Question 4:** verify for

**Solution :** Let

Then

Also,

Now,

`

Now,

Also,

Hence,

**Question 5:** Find the inverse of each of the matrix (if it exist).

**Solution:** Let

Then,

Now,

**Question 6:** Find the inverse of the matrix (if it exists)

**Solution:** Let

Then

Now,

Therefore,

Hence,

**Question 7:** Find the inverse of the matrix (if it exists)

**Solution:** Let

Then,

Now,

Therefore,

Hence,

**Question 8:** Find the inverse of each of the matrix(if it exists)

**Solution:** Let

Then

Now,

**Question 9:** Find the inverse of each of the matrix(if it exists)

**Solution:**

Then,

Now,

Hence,

**Question 10: ** Find the inverse of the matrix (if it exists).

**Solution:** Let

Then, expanding

Now,

Therefore,

Hence,

**Question 11:** Find the inverse of the matrix (if it exists)

**Solution:** Let

Then,

Now,

Therefore,

**Question 12:** Let and verify that

**Solution: ** Let

Then,

Now,

Therefore,

Now for B

Let

Then,

Now,

Then,

Now,

Also,

Then we have,

and

Therefore,

Thus,

From (1) and (2),

Hence, proved.

**Question 13:** If , show that . Hence find .

**Solution:** Let

Therefore,

Now,

Hence, .

Now,

Thus,

**Question 14:** For the matrix , find the numbers and such that .

**Solution:** Let

Therefore,

Now, .

Hence,

Now,

From (1) and (2), we have,

Comparing the corresponding elements of the two matrices, we have:

Also,

Thus, and .

**Question 15: **

show that

Hence, find .

**Solution:** Let

Therefore,

And,

Hence,

Thus,

Now,

Now,

From equation (1) and (2)

**Question 16:** If

verify that

. Hence, find .

**Solution:** Let

Therefore,

And

Now,

Thus,

Now,

Now,

From equations and

**Question 17:** Let be a non-singular square matrix of order . Then is equal to:

(A)

(B)

(C)

(D)

**Solution: the correct option is B.**

Since be a non-singular square matrix of order

Therefore,

Thus, the correct option is B.

**Question 18:** If is an invertible matrix of order 2, the is equal to:

(A)

(B)

(C) 1

(D) 0

**Solution:** Thus, the correct option is B.

Since is an invertible matrix, exists and adj .

As matrix is of order 2, let Then,

And

Now,

Hence,

Hence,

Thus, the correct option is B.