Perform operations on matrices and use matrices in applications.

Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Use matrices to represent and manipulate data, e.g., transformations of vectors.

Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

Add, subtract, and multiply matrices of appropriate dimensions.

Multiply matrices by scalars to produce new matrices.

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.