Question

Suppose \( Q \in M_{3 \times 3}(\mathbb{R}) \) is a matrix of rank 2. Let \( T : M_{3 \times 3}(\mathbb{R}) \to M_{3 \times 3}(\mathbb{R}) \) be the linear transformation defined by

\[ T(P) = QP. \]

Then the rank of \( T \) is ..................

Hint:

The rank of a linear transformation involving matrix multiplication is determined by the rank of the multiplying matrices.

Answer 1

Correct Answer: 6

Step 1: Understand the transformation

$T$ is a linear transformation that left-multiplies any $3 \times 3$ matrix $P$ by the fixed matrix $Q$.

Step 2: Represent $T$ as matrix multiplication

We can vectorize the matrices. For a matrix $P$, we can write it as a vector $\text{vec}(P)$ of dimension 9.

The transformation $T(P) = QP$ can be represented as: $$\text{vec}(QP) = (I \otimes Q)\text{vec}(P)$$

where $\otimes$ denotes the Kronecker product and $I$ is the $3 \times 3$ identity matrix.

Step 3: Find the rank of the transformation

The rank of $T$ equals the rank of the matrix representation $(I \otimes Q)$.

Property of Kronecker product: If $A$ is $m \times n$ with rank $r_A$ and $B$ is $p \times q$ with rank $r_B$, then: $$\text{rank}(A \otimes B) = r_A \cdot r_B$$

Step 4: Apply the property

• $I$ is $3 \times 3$ with rank 3
• $Q$ is $3 \times 3$ with rank 2

Therefore: $$\text{rank}(I \otimes Q) = \text{rank}(I) \times \text{rank}(Q) = 3 \times 2 = 6$$

Answer: 6