= Solution
If the change is purely a rescaling of the physics mark, write $X'=DX$, where $D=\operatorname{diag}(10/3,1,1,1)$. The <sample covariance matrix> transforms as
$$
\Sigma'=D\Sigma D.
$$
The physics <variance> is multiplied by $100/9$, becoming approximately $10044.44$. Each covariance between physics and another subject is multiplied by $10/3$, and all other entries stay unchanged. <Principal component analysis> on this new <sample covariance matrix> maximizes <variance> in the new numerical units, so its <eigenvectors> and <explained variance of a principal component> generally change. Physics receives disproportionately greater influence simply because its unit has changed.
One solution is to convert the physics marks back to the original scale before doing <principal component analysis>. Alternatively, use <principal component analysis on a correlation matrix>: standardize every subject by
$$
Z_j=\frac{X_j-\bar X_j}{s_j},\qquad R_{jk}=\frac{\Sigma_{jk}}{s_js_k},\qquad s_j=\sqrt{\Sigma_{jj}}.
$$
A positive change of units multiplies both a centered variable and its standard deviation by the same factor. Thus it leaves $Z_j$ and the <sample correlation matrix> $R$ unchanged.
\b[Correlation-based principal component analysis is invariant under positive changes of units; covariance-based principal component analysis is not.] Standardizing gives each subject unit <variance>, which is a different weighting choice and can emphasize relatively low-variance subjects. If the revised assessment changes the scores themselves rather than just their units, its <sample correlation matrix> may also change, and mere standardization cannot recover the original analysis.
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