The adjugate matrix of a square matrix A, written adj A, is the transpose of the matrix of cofactors of A. It gives a formula for the inverse: if det A ≠ 0, then A⁻¹ = (1/det A) adj A. For a 2 × 2 matrix [a b; c d], adj A = [d −b; −c a].
The properties of determinants tell you how a determinant changes when you change the matrix. Swapping two rows flips the sign, multiplying a row by k multiplies the determinant by k, and adding a multiple of one row to another changes nothing. Also, det(AB) = det A · det B and det(Aᵀ) = det A.
Cofactor expansion finds the determinant of any square matrix. Pick one row or column, multiply each entry by its cofactor, and add the products. Each cofactor is a smaller determinant with a + or − sign, so a 4 × 4 determinant becomes 3 × 3 ones. Expanding along the line with the most zeros saves the most work.
An LU decomposition writes a square matrix A as A = LU, where L is lower triangular with 1s on its diagonal and U is upper triangular. U is the echelon form from Gaussian elimination, and L stores the multipliers used. To solve Ax = b, solve Ly = b by forward substitution, then Ux = y by back substitution.
The invertible matrix theorem says that for a square n x n matrix A, many statements are all true or all false together: A is invertible, A has n pivots, Ax = 0 has only the trivial solution, the columns of A are independent and span R^n, and det A is not 0. One check settles them all.
An elementary matrix is the matrix you get by doing one row operation to an identity matrix. Multiplying a matrix A on the left by an elementary matrix E does that same row operation to A, so EA is the result. Every elementary matrix is invertible, and its inverse undoes the row operation.
The transpose of a matrix A, written A^T, is the matrix made by turning the rows of A into columns: row i of A becomes column i of A^T, so the entry in row i, column j of A^T is the entry in row j, column i of A. An m x n matrix has an n x m transpose.
The composition of linear transformations S ∘ T applies T first and then S, so (S ∘ T)(x) = S(T(x)). If T has standard matrix A and S has standard matrix B, then S ∘ T is also linear and its standard matrix is the product BA. The matrix of the transformation applied first goes on the right.
One-to-one and onto transformations describe how a linear transformation T(x) = Ax moves vectors. T is one-to-one when different inputs always give different outputs, and onto when every vector in the codomain R^m is an output. Row reduce A: T is one-to-one when every column has a pivot and onto when every row has a pivot.
The standard matrix of a linear transformation T: ℝⁿ → ℝᵐ is the m × n matrix A = [T(e1) … T(en)], whose columns are the images of the standard basis vectors e1, …, en. It is the one matrix with T(x) = Ax for every x, so you find it by applying T to each ei.