Linear Equations & Matrices

Which letters represent rows and columns in a matrix?

m = rows and n = columns

What is a homogeneous system of linear equations?

If all the equations equal 0

How do we denote the set of all m x n matrices?

M_mn(R)

What does A = [a_ij] represent?

The entry written in the ith row and jth column of a matrix, written a_{i, j} = A where A is the entry

What are the properties of matrix multiplication?

• Closed under matrix multiplication

• Associative

• Distributive

• There exists an identity

• Not commutative

What is a diagonal matrix?

An n x n marix when a_{i, j} = 0 when i doesn’t equal j

What is an upper triangular matrix?

An n x n matrix with a_{i, j} = 0 for all i > j

What is a lower triangular matrix?

An n x n matrix with a_{i, j} = 0 for all j > i

What are singular and non-singular matrices?

A singular matrix is invertible (it has an inverse) and a non-singular matrix is non invertible (no inverse)

What is an augmented matrix?

What are the elementary row operations?

• Interchange/switch 2 rows

• Multiply a row by a non-zero constant

• Add a multiple of one row to another

When is a matrix in echelon form?

• All rows consisting of only 0 are at the bottom of the matrix

• The first non-zero number in any row is 1

• Successive non-zero rows begin with more 0s left of the 1 than the rows above

When is a matrix in reduced echelon form (REO)?

If it is in echelon form and the first non-zero entry in each row is the only non-zero entry in the column

What is the strategy for Gaussian elimination?

• Form the augmented matrix

• Use EROs to reduce the augmented matrix to echelon form

• Solve the problem given in echelon form by back substitution or reducing to reduced echelon form

What are consistent and non-consistent equations?

Equations with a unique solution or infinite solutions are consistent and equations with no solution are inconsistent