18. Properties of Determinants

Overview

The lecture builds the determinant from three defining rules: the identity has determinant 1, swapping two rows reverses the sign, and the determinant is linear in each row separately. These rules yield practical consequences, including zero determinants for repeated or zero rows, invariance under row replacement, and the pivot-product method for computing determinants and testing invertibility. The lecture also establishes determinant rules for products, inverses, scaling, and transposes.

Key takeaways

The determinant is defined by three rules: det(I) = 1, a row swap changes its sign, and it is linear in each row when the other rows are fixed.
Elimination row replacements do not change the determinant, while each row exchange changes its sign; therefore determinant computation reduces to multiplying triangular pivots and tracking swap parity.
A square matrix is invertible exactly when its determinant is nonzero, because elimination yields nonzero pivots for an invertible matrix and a zero row for a singular one.
The determinant is multiplicative, det(AB) = det(A)det(B), which implies det(A⁻¹) = 1/det(A) for invertible A.
For an n×n matrix, multiplying every entry by a scalar c multiplies the determinant by cⁿ, since the scalar can be factored from each of the n rows.
Transposition preserves the determinant, so determinant properties for rows also yield corresponding properties for columns.

Chapters

0:00 Determinants Begin with Three Rules and the Identity Matrix

A determinant is a single number associated with each square matrix, and a nonzero determinant indicates invertibility.
The first defining rule is det(I) = 1; swapping two rows reverses the determinant's sign.
Applying row swaps to the identity gives permutation matrices, whose determinants are +1 or -1 according to whether the number of swaps is even or odd.
For a 2×2 matrix, the familiar determinant formula is ad − bc.

7:10 Row-by-Row Linearity Defines the Third Determinant Rule

Scaling one row by a factor T scales the determinant by T, allowing that factor to be pulled outside the determinant.
If one row is a sum of two rows, with all other rows held fixed, the determinant splits into the sum of two determinants.
The determinant is linear in each row separately, not generally linear in the entire matrix: det(A + B) need not equal det(A) + det(B).
Linearity in rows other than the first follows by swapping rows, applying the rule, and swapping them back.

11:17 Equal Rows Force the Determinant to Be Zero

If two rows are equal, swapping them leaves the matrix unchanged but also reverses the determinant's sign.
The only value equal to its own negative is zero, so any matrix with repeated rows has determinant zero.
A matrix with equal rows has rank less than its size and is therefore singular.

14:30 Elimination Row Replacements Preserve the Determinant

Replacing row k with row k minus a multiple of another row leaves the determinant unchanged.
Row linearity splits the replacement into two determinant terms; the term with duplicate rows vanishes by the equal-row rule.
A complete row of zeros gives determinant zero, which also follows from row linearity by scaling that row by zero.
These rules let elimination simplify a matrix while tracking determinant changes from row swaps.

22:10 Triangular Matrices Have Determinants Equal to Pivot Products

For an upper-triangular matrix with diagonal pivots d₁ through dₙ, the determinant is d₁d₂⋯dₙ.
When all pivots are nonzero, row replacements eliminate the entries above the diagonal without changing the determinant.
For a diagonal matrix, factoring each dᵢ out of its row leaves the identity, whose determinant is 1.
If a pivot is zero, elimination produces a zero row, so the determinant is zero.

28:00 Elimination Connects Determinants to Invertibility

A matrix is singular exactly when elimination produces a zero row, corresponding to a zero determinant.
For an invertible matrix, elimination produces nonzero pivots, and their product gives the determinant if no row swaps occur.
Each row exchange changes the sign, so the final pivot product must be adjusted by the parity of the swaps.
For a 2×2 matrix, elimination gives pivots a and d − cb/a; multiplying them recovers ad − bc when a is a valid first pivot.

33:00 Products, Inverses, and Scaling Follow Determinant Rules

The determinant is multiplicative: det(AB) = det(A)det(B), even though determinants are not generally additive.
Since A⁻¹A = I, an invertible matrix satisfies det(A⁻¹) = 1/det(A).
Squaring a matrix squares its determinant: det(A²) = det(A)².
Scaling every entry of an n×n matrix by 2 multiplies its determinant by 2ⁿ, because a factor of 2 comes from each of the n rows.

41:36 Transposition Preserves Determinants and Extends Row Rules to Columns

The transpose rule is det(Aᵀ) = det(A); for a 2×2 matrix, both the original and transposed matrices have determinant ad − bc.
Because transposition exchanges rows and columns, a zero column forces determinant zero just as a zero row does.
Swapping two columns reverses the determinant's sign, by transposing, swapping the corresponding rows, and transposing back.
The lecture begins a proof using A = LU and Aᵀ = UᵀLᵀ, together with the multiplicative rule and triangular-matrix determinants.

46:00 Even and Odd Row Swaps Make the Determinant Well-Defined

An even number of row exchanges preserves the sign, while an odd number reverses it.
A permutation can be represented by different sequences of row swaps, so the parity of those sequences must agree.
The parity fact ensures the determinant is consistently defined by the three original rules and supports the derived properties.

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