(a) Give an example of a non-diagonal 2 ×2 matrix for which A^2=I. (b) In general, if A^2=I, sho

step 1 In this problem we will need to use the concept of matrix multiplication. Now what we need to do

(a) Give an example of a non-diagonal 2 ×2 matrix for which...

(a) Give an example of a non-diagonal $2 \times 2$ matrix for which $A^2=\mathrm{I}$. (b) In general, if $A^2=\mathrm{I}$, show that $\operatorname{det} A= \pm 1$. (c) If $A^2=A$, what can you say about $\operatorname{det} A$ ?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Determinant Properties

The determinant is a scalar attribute of a matrix that provides key insights about the matrix, such as invertibility and volume scaling in linear transformations. In the context of involutory matrices (where A² = I), taking the determinant on both sides yields (det A)² = det(I) = 1, which implies that det A must be either +1 or -1. This fundamental property links the algebraic behavior of the matrix to its geometric interpretation.

Idempotent Matrix

An idempotent matrix is one that satisfies the relation A² = A. This property implies that the eigenvalues of an idempotent matrix are either 0 or 1, since any eigenvalue ? must satisfy the equation ?² = ?. As a result, the determinant of an idempotent matrix, which is the product of its eigenvalues, is either 0 or 1, reflecting whether the matrix is singular or non-singular.

Involutory Matrix

An involutory matrix is a square matrix that is its own inverse, meaning that when it is multiplied by itself, the result is the identity matrix. This property is expressed as A² = I. An important characteristic of involutory matrices is that they have eigenvalues that are either +1 or -1, which is closely related to the behavior of their determinants and other matrix properties.

Transcript

Please give Ace some feedback