= Solution
Treat the displayed entries as the lower triangle of a symmetric <sample covariance matrix> $S$, in the order mechanics, vectors, algebra, analysis, statistics. Compute the <precision matrix> $K=S^{-1}$. For a <multivariate normal distribution>, conditioning on all remaining coordinates gives
$$
\widehat{\operatorname{Var}}(Y_j\mid Y_{-j})=\frac1{K_{jj}},\qquad
\widehat{\operatorname{Corr}}(Y_i,Y_j\mid Y_{-(i,j)})
=-\frac{K_{ij}}{\sqrt{K_{ii}K_{jj}}}.
$$
These identities follow by fixing the other coordinates in the quadratic form of the normal density and completing the square. For the second identity, the conditional <precision matrix> for the retained pair is its $2\times2$ principal block; inverting that block gives the negative off-diagonal entry divided by the square root of the two diagonal entries.
Thus the two quantities to evaluate are
$$
\boxed{\frac1{K_{11}S_{11}}\simeq0.62,\qquad
-\frac{K_{54}}{\sqrt{K_{55}K_{44}}}\simeq0.25.}
$$
An equivalent procedure uses the <Schur complement covariance>: regress mechanics on the other four variables, or regress analysis and statistics separately on the remaining three. The first residual <variance> is $S_{11}-S_{1,-1}S_{-1,-1}^{-1}S_{-1,1}$; the second result is the <partial correlation> of the two residuals. No arithmetic is required to specify this method.
The model qualification is important: a covariance matrix alone determines linear-regression residual variances and <partial correlations>. Identifying them with conditional quantities independent of the observed conditioning values uses the <multivariate normal> model, or another model with the same conditional moments; it does not follow for arbitrary distributions.
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