GATE CS 2022 Q10 — Linear Algebra: Matrices, Determinants, System of Linear Equations

The trace of a square matrix is defined as the sum of its diagonal elements. A fundamental property of the trace is its commutativity under multiplication, known as the cyclic property: for any two matrices $X_{p \times q}$ and $Y_{q \times p}$, we have:

Proof of the Property:

Let $X$ be a $p \times q$ matrix and $Y$ be a $q \times p$ matrix.

• The product $XY$ is a square matrix of size $p \times p$. Its trace is: $$\text{tr}(XY) = \sum_{i=1}^{p} (XY)_{ii} = \sum_{i=1}^{p} \sum_{j=1}^{q} X_{ij} Y_{ji}$$
• The product $YX$ is a square matrix of size $q \times q$. Its trace is: $$\text{tr}(YX) = \sum_{j=1}^{q} (YX)_{jj} = \sum_{j=1}^{q} \sum_{i=1}^{p} Y_{ji} X_{ij}$$

Since scalar multiplication is commutative ($X_{ij} Y_{ji} = Y_{ji} X_{ij}$) and finite summations can be reordered, we have:

Thus, $\text{tr}(XY) = \text{tr}(YX)$ always holds, even if the dimensions of $XY$ ($p \times p$) and $YX$ ($q \times q$) are different.

Evaluating the Statements:

• Statement 1: For $A_{m \times n}$ and $B_{n \times m}$, both $AB$ (size $m \times m$) and $BA$ (size $n \times n$) are square matrices. By the cyclic property, $\text{tr}(AB) = \text{tr}(BA)$. Hence, Statement 1 is correct.
• Statement 2: For $C_{n \times n}$ and $D_{n \times n}$, both $CD$ and $DC$ are square matrices of size $n \times n$. By the cyclic property, $\text{tr}(CD) = \text{tr}(DC)$. Hence, Statement 2 is correct.

Therefore, both Statement 1 and Statement 2 are correct.