Determinants: questions for repetition
All formulas for calculating determinants are complex. There are two ways to study determinants: one is to start with the explicit expressions (the Leibniz formula or Laplace expansion) and the other is to start with simple functional properties (called Axioms 1-3 here), then develop more advanced ones (multilinearity) and, finally, derive explicit formulas. I prefer to see ideas and follow the second way.
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Formulate Axioms 1-3 and make sure that you understand the motivation: 1) multiplying one of the equations of the system
by a nonzero number
does not impact solvability of the system, 2) similarly, adding one equation of the system to another does not affect solvability and 3) the system
is trivially solvable.
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Interpret in terms of system solvability properties I-III and prove them.
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Prove that the determinant is a multilinear antisymmetric function of rows.
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Define a permutation matrix and give an example of calculating its determinant using Axioms 1-3 and Properties I-VI.
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Show what pre-multiplication by a permutation matrix does to a matrix
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Prove that a permutation matrix is an orthogonal matrix.
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Derive the Leibniz formula. At this point, the way multilinearity (in rows) and permutation matrices are used should be absolutely obvious. If they are not, start from Question 1.
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Explain the different-rows-different-columns (cross-out) rule.
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Prove multiplicativity of determinants.
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Prove multilinearity in columns.
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Prove that transposition does not change determinants.
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Let
be the
-th column of
and define the row-vector
by
Why is this definition correct? Prove that
for any
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Using Question 12, derive Cramer's rule for solving the system
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Using Question 12, prove the invertibility criterion.
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Derive the Laplace expansion.
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Using Questions 14 and 15, derive the explicit formula for
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