Question

Find the value of y for which the matrix is equal to its own inverse (a) 3y (b) y²-2-34

Answer 1

(a) The value of y for which the matrix is equal to its own inverse is y = 1/√3. Thus the correct option is A.

Step-by-step explanation:

For a matrix to be equal to its own inverse, let's denote the matrix as A. The condition for A to be its own inverse is A =
`A^{(-1)`. In terms of matrices, this means that the product of A and its inverse equals the identity matrix, i.e., A * A = I.

Given matrix A = 3y, to find its inverse, we'll solve A * A = I for A =
`A^{(-1)`:

A * A = I

(3y) * (3y) = I

9y² = I

For a 2x2 identity matrix, I = [1 0; 0 1]. Therefore, 9y² should equal the identity matrix:

9y² = [1 0; 0 1]

The identity matrix here implies that 9y² should equal 1, as it is the only value in the diagonal of the identity matrix. So:

9y² = 1

y² = 1/9

y = ±√(1/9)

y = ±1/3

However, in the context of matrix inverses, we consider the positive square root as negative values may lead to scaling of the matrix by -1, not true inversion. Thus, y = 1/√3.

This value of y (y = 1/√3) ensures that the given matrix, 3y, is indeed equal to its own inverse.Thus the correct option is A.