Matrices and Determinants

Expresses every square matrix uniquely as the sum of a symmetric and a skew-symmetric matrix.

Gives the signed minor associated with the element in row i and column j.

Evaluates the determinant of a square matrix along any chosen row or column.

Defines the adjoint of a matrix as the transpose of its cofactor matrix.

Computes the adjoint of a 2x2 matrix by swapping diagonal elements and negating off-diagonal elements.

Gives the multiplicative inverse of any non-singular square matrix.

Direct formula for computing the inverse of an invertible 2x2 matrix.

Solves the matrix equation AX = B when the coefficient matrix A is non-singular.

Determines whether the non-homogeneous linear system AX = B has a unique solution, infinitely many, or none.

Solves a system of n linear equations in n variables using determinants of coefficient matrices.

Scales the determinant by k to the power of the order n when multiplying matrix A by scalar k.

From: Root fact (alternating). Same for two columns.

From: Swap the equal rows: Δ = −Δ, so Δ = 0; a factor k comes out first.

From: Linear in that row: take the factor 0 out.

From: Root fact (linear in each row).

From: Root fact (linear in each row). Split ONE row at a time.

From: Split the row (sum card) → second determinant has proportional rows → 0.

From: It is the expansion of a determinant with two equal rows. Gives A·adj A = |A|I.

From: Expand down the first column again and again. Diagonal and scalar matrices are special cases; |I| = 1.

From: Zero when a = b (two equal columns), so (a − b) is a factor; likewise (b − c), (c − a); compare degrees.

From: C₁ → C₁ + C₂ + C₃ takes out (a + b + c).

From: Δ(x) is a polynomial in x; Δ(α) = 0 (often two equal rows at x = α) ⇒ remainder theorem. Use: factorise without expanding.

From: Factor theorem: x = y, y = z, z = x make it 0; degree 4, so one more symmetric factor k(x + y + z); x, y, z = 0, 1, 2 gives k = 1.

From: Root fact.

From: |A| = |Aᵀ| = |−A| = (−1)ⁿ|A| = −|A|.

From: Order 2: |[0 a; −a 0]| = a². In general |A| = (Pfaffian)².

From: |A||Aᵀ| = |I| ⇒ |A|² = 1.

States that the determinant of the product of two square matrices equals the product of their determinants.

Gives the inverse of a product of invertible matrices by reversing the factor order.

States that the determinant of the inverse matrix is the reciprocal of the matrix determinant.

From: |AB| = |A||B| repeated.

From: Both are |A||B|.

From: A⁻¹A = I says A is the inverse of A⁻¹.

From: Reversal law with every factor equal to A.

From: Transpose AA⁻¹ = I: (A⁻¹)ᵀAᵀ = I.

From: (kA)(¹⁄ₖA⁻¹) = I.

From: Multiply: each diagonal entry becomes 1.

Gives the scalar matrix product obtained by multiplying any square matrix by its adjoint.

Gives the determinant of the adjoint of an n x n square matrix.

From: adj X = |X| X⁻¹ and the reversal law.

From: Product card with every factor A.

From: Cofactors of Aᵀ are the transposed cofactors of A.

From: Each cofactor is an (n−1)×(n−1) determinant: k comes out n − 1 times.

From: Put adj A in place of A in the root: adj A · adj(adj A) = |A|ⁿ⁻¹ I; multiply by A.

From: Apply |adj X| = |X|ⁿ⁻¹ twice.

From: Each adj raises the power by a factor n − 1.

From: adj A = |A|A⁻¹; replace A by A⁻¹.

From: Each cofactor is the product of the other two diagonal entries.

From: Root with |A| = 0.

Gives the transpose of a matrix product by multiplying the individual transposes in reverse order.

From: Reversal law + (Aᵀ)ᵀ = A. A may be any m×n matrix.

From: (BᵀAB)ᵀ = BᵀAᵀB.

From: (AB)ᵀ = BᵀAᵀ = BA.

From: Transpose each: (AB)ᵀ = BA for both cases.

From: (Aᵏ)ᵀ = (Aᵀ)ᵏ = (−1)ᵏAᵏ.

From: aᵢᵢ = −aᵢᵢ.

From: XᵀAX is 1×1, so it equals its transpose XᵀAᵀX = −XᵀAX.

From: Diagonal + upper triangle; skew has a zero diagonal. Count of such matrices with entries from k values: k^{n(n+1)/2}.

From: Expand keeping the order.

From: Expand keeping the order.

From: Commuting matrices behave like numbers. Typical use: (aI + N)ⁿ with N nilpotent.

From: Powers of ONE matrix always commute.

From: Diagonal entries add.

From: Both are Σᵢ Σⱼ aᵢⱼ bⱼᵢ.

From: Aⁿ = Aⁿ⁻² · A² = Aⁿ⁻¹ = … = A.

From: Binomial (I and A commute) with Aʳ = A: Σ C(n, r) for r ≥ 1 is 2ⁿ − 1. Also I − A is idempotent.

From: A · A = I.

From: Multiply: (I − N)(I + N + … + Nᵏ⁻¹) = I − Nᵏ = I. Also |N| = 0.

From: Definition read as an inverse.

Condition for the homogeneous linear system AX = O to have non-zero (infinitely many) solutions.

From: Completes the consistency card: here the test is inconclusive (e.g. solve two equations, check the third).

From: X = A⁻¹O = O.

Relates the square of any 2x2 matrix to its trace, determinant, and identity matrix.

From: (A − λI)X = O has X ≠ O only when A − λI is singular.

From: Coefficients of |A − λI|; put λ = 0 for the product.

From: Apply the matrix to AX = λX; Aᵀ: |Aᵀ − λI| = |A − λI|.

From: Every square matrix satisfies its own characteristic equation (the 2×2 card is the case n = 2). Use: powers and A⁻¹ from p(A) = O.