Step 1: Understanding the Question
We need to first perform the matrix addition of a \(3 \times 3\) identity matrix and a \(3 \times 3\) null (zero) matrix, and then find the rank of the resulting matrix.
Step 2: Key Formula or Approach
Key matrix properties needed:
Identity Matrix (\(I\)): A square matrix with 1s on the main diagonal and 0s elsewhere.
Null Matrix (\(O\)): A matrix where all elements are 0.
Matrix Addition Property: For any matrix \(A\), \(A + O = A\).
Rank of a Matrix: The number of linearly independent rows (or columns) in the matrix. The rank of an \(n \times n\) identity matrix, \(I_n\), is \(n\).
Step 3: Detailed Explanation
First, let's write the matrices:
The \(3 \times 3\) identity matrix is:
\[
I_3 = \begin{pmatrix} 1 & 0 & 0
0 & 1 & 0
0 & 0 & 1 \end{pmatrix}
\]
The \(3 \times 3\) null matrix is:
\[
O = \begin{pmatrix} 0 & 0 & 0
0 & 0 & 0
0 & 0 & 0 \end{pmatrix}
\]
Now, add the two matrices:
\[
I_3 + O = \begin{pmatrix} 1 & 0 & 0
0 & 1 & 0
0 & 0 & 1 \end{pmatrix} + \begin{pmatrix} 0 & 0 & 0
0 & 0 & 0
0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0
0 & 1 & 0
0 & 0 & 1 \end{pmatrix} = I_3
\]
The resulting matrix is the identity matrix \(I_3\). The rank of \(I_3\) is the number of its linearly independent rows, which is 3.
Step 4: Final Answer
The rank of the resulting matrix is 3.