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

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Question 5:

If , then show that is equal to zero.

Interchanging rows and columns, we get

Taking ’ common from and , we get

Question 6:

Prove that , where A is an invertible matrix.

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.,

Question 7:

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

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.

Expanding along , we get

triangle ABC is isoceles.

Question 9:

Show that if the determinant then

we get

or

or

or

or

or (Why ?).