Question

What is the rank of a \(3 \times 3\) identity matrix added to a \(3 \times 3\) null matrix?

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(3\)

Hint:

The rank of an identity matrix \(I_n\) is always \(n\), because all rows and columns are linearly independent.

Answer 1

The Correct Option is D

Concept:
The identity matrix \(I_n\) is a square matrix with ones on the main diagonal and zeros elsewhere. The null matrix (or zero matrix) has all its elements equal to zero. Key properties:

• Adding a null matrix to any matrix does not change the matrix. \[ A + O = A \]
• The rank of the identity matrix \(I_n\) is \(n\) because all rows (and columns) are linearly independent.

Step 1: Add the matrices.
Let the identity matrix be \[ I_3 = \begin{bmatrix} 1 & 0 & 0
0 & 1 & 0
0 & 0 & 1 \end{bmatrix} \] The null matrix is \[ O = \begin{bmatrix} 0 & 0 & 0
0 & 0 & 0
0 & 0 & 0 \end{bmatrix} \] Adding them: \[ I_3 + O = I_3 \]
Step 2: Find the rank of the resulting matrix.
Since the resulting matrix is still the identity matrix \(I_3\), all its rows (and columns) are linearly independent. \[ \text{Rank}(I_3) = 3 \] \[ \therefore \text{The rank of the matrix is } 3. \]