Chapter 2 Matrices and vectors

This section is about matrices and vectors. We introduce the basics of matrix algebra and look at applications to linear systems, including row reduced echelon form. We will define the determinant and discuss its meaning and methods of computing it.

Learning objectives for this section:

• Know the definitions of matrix, vector, transpose, matrix multiplication, matrix addition, scalar multiplication, inverse matrix, determinant, row operation, row reduced echelon form, characteristic polynomial, eigenvalue, eigenvector.
• Compute sums, products, transposes, and scalar multiples of matrices.
• Find determinants, and know their relationship with invertibility.
• Choose appropriate methods to determine whether a matrix is invertible, and find the inverse if it exists.
• Understand when two matrices can be multiplied, and the motivation for the definition of matrix multiplication.
• Be able to find the row reduced echelon form of a matrix and use it to identify invertible matrices, compute matrix inverses when they exist, and solve systems of linear equations.
• Be able to compute eigenvalues and eigenvectors using the characteristic polynomial.