# NCERT Class 12-Mathematics: Exemplar Chapter – 4 Determinants Part 3 (For CBSE, ICSE, IAS, NET, NRA 2023)

Doorsteptutor material for CBSE/Class-12 is prepared by world's top subject experts: get questions, notes, tests, video lectures and more- for all subjects of CBSE/Class-12.

**Question 5**:

If , then show that is equal to zero.

**Answer**:

Interchanging rows and columns, we get

Taking ′ common from and , we get

**Question 6**:

Prove that , where A is an invertible matrix.

**Answer**:

Since A is an invertible matrix, so it is non-singular.

We know that . But . So i.e.. is invertible matrix.

Now we know that .

Taking transpose on both sides, we get

Hence is inverse of , i.e.. ,

## Long Answer (L. A.)

**Question 7**:

If is a root of , then find the other two roots

**Answer**:

Applying , we get

Taking common from , we get

Applying , we get

Expanding along ,

. Thus, implies

**Question 8**:

In a triangle , if

then prove that is an isosceles triangle.

**Answer**:

Expanding along , we get

triangle ABC is isoceles.

**Question 9**:

Show that if the determinant then

**Answer**:

we get

or

or

or

or

or (Why?) .