Question: If A is a 2x2 matrix such that A^2 = I, where I is the identity matrix, then which of the following is true?
Options:
A is invertible
A is singular
A is a zero matrix
A is a diagonal matrix
Correct Answer: A is invertible
Solution:
Since A^2 = I, A is invertible because the inverse of A is A itself.
If A is a 2x2 matrix such that A^2 = I, where I is the identity matrix, then whi
Practice Questions
Q1
If A is a 2x2 matrix such that A^2 = I, where I is the identity matrix, then which of the following is true?
A is invertible
A is singular
A is a zero matrix
A is a diagonal matrix
Questions & Step-by-Step Solutions
If A is a 2x2 matrix such that A^2 = I, where I is the identity matrix, then which of the following is true?
Step 1: Understand what A^2 = I means. This means that when you multiply matrix A by itself, you get the identity matrix I.
Step 2: Recall what the identity matrix is. For a 2x2 matrix, the identity matrix I looks like this: [[1, 0], [0, 1]].
Step 3: Recognize that if A^2 = I, then A is its own inverse. This means that A multiplied by A gives you the identity matrix.
Step 4: Understand the concept of an invertible matrix. A matrix is invertible if there exists another matrix that, when multiplied with it, gives the identity matrix.
Step 5: Since A multiplied by A gives the identity matrix, it means that A can be inverted by itself. Therefore, A is invertible.
Matrix Inversion – Understanding that if A^2 = I, then A is its own inverse, meaning A is invertible.
Identity Matrix Properties – Recognizing that the identity matrix I has properties that relate to matrix multiplication and inversion.
Eigenvalues and Eigenvectors – Knowing that if A^2 = I, the eigenvalues of A can only be ±1.
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