- The identity matrix \(I\) is the multiplicative identity for matrices, satisfying \(AI=IA=A\).
- It is a square matrix with ones on the main diagonal and zeros elsewhere.
- For a \(3 \times 3\) matrix, the identity matrix is \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\).
- The final answer is \(\boxed{\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}}\).
Explanation
- Definition of the Identity Matrix
The identity matrix, denoted by \(I\), is a square matrix that acts as the multiplicative identity in matrix algebra. For any square \(n \times n\) matrix \(A\), the identity matrix \(I_n\) satisfies the property \(AI_n = I_nA = A\). This means that multiplying a matrix by the identity matrix leaves the matrix unchanged. We are tasked with identifying the \(3 \times 3\) identity matrix from the provided options.
- Identifying the Identity Matrix Structure
A \(3 \times 3\) identity matrix is defined as a square matrix where all elements on the main diagonal (from top-left to bottom-right) are equal to 1, and all other elements are equal to 0. Mathematically, the entries \(I_{ij}\) of the identity matrix are given by:
\[I_{ij} = \begin{cases} 1 & \text{if } i = j \ 0 & \text{if } i
eq j \end{cases}\]
This results in the matrix:
\[I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]
- Evaluating the Options
Let us examine the given options:
- Option 1: \(\begin{bmatrix} 1 & 1 & 1 \end{bmatrix}\) is a \(1 \times 3\) row vector, not a \(3 \times 3\) matrix.
- Option 2: \(\begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1 \end{bmatrix}\) is a \(3 \times 3\) matrix consisting of all ones, which does not act as an identity matrix under multiplication.
- Option 3: \(\begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}\) is a \(3 \times 1\) column vector, not a \(3 \times 3\) matrix.
- Option 4: \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\) matches the required structure of the \(3 \times 3\) identity matrix.
Therefore, the correct identity matrix is \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\).
Examples
The identity matrix is fundamental in computer graphics and game development. For instance, when applying transformations like rotation, scaling, or translation to 3D objects, these transformations are represented by matrices. The identity matrix represents a "no-operation" transformation, meaning if you multiply an object's position matrix by the identity matrix, its orientation and location remain completely unchanged. This serves as the baseline from which all other dynamic changes are calculated.