# NCERT Class 12-Mathematics: Exemplar Chapter –4 Determinants Part 3

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**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 ?).