Matrices And Determinants

Step 1: Understanding the matrix powers.
The matrix

A

is a 2x2 matrix with the property:

A%5E2%20%3D%20%5Cbegin%7Bpmatrix%7D%200%20%26%201%20%5C%5C%20-1%20%26%200%20%5Cend%7Bpmatrix%7D%5E2%20%3D%20%5Cbegin%7Bpmatrix%7D%20-1%20%26%200%20%5C%5C%200%20%26%20-1%20%5Cend%7Bpmatrix%7D%20%3D%20-I

where

I

is the identity matrix. Step 2: Generalizing for $A%5E%7B2n%7D$ .
Since

A%5E2%20%3D%20-I

, for any integer

n

, we have:

A%5E%7B2n%7D%20%3D%20(-I)%5En%20%3D%20(-1)%5En%20I

Thus,

A%5E%7B2n%7D

is either

I

-I

, depending on whether

n

is even or odd. Step 3: Conclusion.
The matrix

A%5E%7B2n%7D

has the form

%5Cbegin%7Bpmatrix%7D%20%5Cpm1%20%26%200%20%5C%5C%200%20%26%20%5Cpm1%20%5Cend%7Bpmatrix%7D

. The correct answer is (2).

Step 1: Properties of cube roots of unity.
The cube roots of unity satisfy the equation $%5Comega%5E3%20%3D%201$ , and the properties are: $1%20%2B%20%5Comega%20%2B%20%5Comega%5E2%20%3D%200%20%5Cquad%20%5Ctext%7Band%7D%20%5Cquad%20%5Comega%5E2%20%3D%20%5Comega%5E%7B-1%7D$ Step 2: Simplifying the determinant.
The given matrix is of the form: $%5Cbegin%7Bvmatrix%7D%20%5Comega%5E2%20%26%20%5Comega%20%26%20%5Comega%5E2%20%5C%5C%20%5Comega%5E2%20%26%20%5Comega%20%26%20%5Comega%5E2%20%5C%5C%20%5Comega%5E2%20%26%20%5Comega%20%26%20%5Comega%5E2%20%5Cend%7Bvmatrix%7D$ Since the rows are identical, the determinant will be 0. Thus, the value of the determinant is $0$ .
Step 3: Conclusion.
The correct answer is (3) 0.

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Matrices And Determinants

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs