Matrices

State the dimension of a matrix and whether it is square, recognise column and row vectors as matrices, and name any entry by its row and column.

Add and subtract matrices of the same dimension, multiply a matrix by a scalar, and evaluate linear combinations of matrices.

Learn how to transpose a matrix or vector, find the dimension and entries of a transpose, and simplify transposes of sums, differences and scalar multiples.

Multiply a matrix by a vector row by row, decide when the product is defined and how many entries it has, and write it as a linear combination of the matrix's columns.

Learn to multiply two matrices one column at a time, decide when a product is defined and what dimension it has, and tell which rearrangements of a product leave it unchanged.

Learn how to transpose a product of matrices, and how to write dot products as matrix products to rearrange them.

Understand the identity matrix as the square matrix with ones on the diagonal and zeros elsewhere, which leaves any matrix or vector unchanged when multiplied, analogous to multiplying by one in arithmetic.

Learn how to compute the determinant of a square matrix using specific formulas and cofactor expansion, and interpret its value as indicating both invertibility and the signed area (or volume) scaled by the matrix.

Understand how the column space of a matrix defines the set of possible outputs $A\vec{x}$, how the rank measures the number of linearly independent columns, and how these concepts determine whether a system $A\vec{x} = \vec{b}$ has solutions, how many solutions, and whether a square matrix is invertible.

Learn how the null space of a matrix captures all solutions to $A\vec{x} = \vec{0}$, how it determines the structure of all solutions to $A\vec{x} = \vec{b}$, how to find a basis for the null space by solving a homogeneous system, and how the rank-nullity theorem links the number of independent columns to the number of free variables.

See how multiplying a vector by a matrix performs a linear transformation, such as scaling, rotating or reflecting that vector, and learn to interpret matrix multiplication as reshaping the entire coordinate grid in a consistent, structured way.

Understand the definition of orthogonal matrices, recognise their appearance in rotation and reflection matrices, and see how these matrices preserve vector lengths, dot products, and geometric structure by satisfying $Q^T Q = I$ and having inverses equal to their transposes.