Subjects Linear Algebra

Trace Cf Minus 1 D90153

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Trace Cf Minus 1 D90153


1. The problem is to find the trace of the matrix $CF - I$, where $C$ and $F$ are matrices and $I$ is the identity matrix. 2. Recall the definition of the trace: the trace of a matrix is the sum of its diagonal elements. 3. The trace of a difference of matrices is the difference of their traces: $$\text{trace}(CF - I) = \text{trace}(CF) - \text{trace}(I).$$ 4. The trace of the identity matrix $I$ of size $n \times n$ is $n$ because all diagonal elements are 1. 5. Therefore, to find $\text{trace}(CF - I)$, you need to compute $\text{trace}(CF)$ and subtract $n$. 6. Without explicit matrices $C$ and $F$, the answer is expressed as $$\text{trace}(CF - I) = \text{trace}(CF) - n.$$