Engng Math 145 Matrices

Coefficient and Augmented Matrix

Augmented is like coefficient but with coefficients to right of equal sign too.

Row echelon form

1. All zero rows are at bottom

2. In each non-zero row, each leading entry is in a column to the left of any leading entries below it.

Reducing to REF is called Guassian elimination.

Elementary row operations

Only do one ERO at a time.

• Multiply rows by constant

• Swapping rows

• Adding or subtracting rows from each other

Parameter variables

Used when REF matrix has a row with >1 non-zero elements OR zero row equal to zero, so infinitely many solutions.

Process: All non-pivot variables must be set to a parameter variable (s, t, u, v), and pivot variables must be solved for in terms of the parameter variables.

eg. 2x + 3y = 4 → x = 2 - (3s) / 2, where y = s

No real solutions & infinitely many solutions

Has zero row equal to non-zero. eg. 0 = 5

Has zero row equal to zero. eg. 0 = 0. This is because some rows are scalar multiples of each other.

Reduced row echelon form

1. Is in REF

2. Every non-zero row’s leading entry is a 1 and there are no other non-zero entries in the leading entry’s column.

Reducing to RREF is called Gauss-Jordan elimination.

Diagonal & scalar matrix

Diagonal matrix is square matrix in which all non-diagonal entries are 0.

Scalar matrix is diagonal matrix where all diagonal entries are equal to each other, but not 0.

Upper and lower triangular matrix

Square matrix where all entries below/above diagonal are zero (opposite to it’s name).

n x n Identity matrix

Square matrix with dimension n.

All non-diagonal entries are zero and diagonal entries are 1.

Mathematically equal to 1.

Equal matrices and zero matrices

Same size and all entries equal.

Zero matrix has only zero entries.

Matrix addition/subtraction and scalar multiplication

Can add matrices if same size arithmetically.

For subtraction use A - B = A + (-1)B

Multiply each entry by scalar.

Matrix multiplication

Dot multiply rows of first matrix with columns of second matrix.

Order matters.

Transpose matrix and symmetric

A^T : Rotate A clockwise 90^o, then mirror horizontally.

Symmetric when A = A^T

Transpose rules

(A + B)^T = A^T + B^T

(A^T)^T = A

(AB)^T = B^TA^T

(kA)^T = kA^T

(A^r)^T = (A^T)^r

For every square matrix A, A + A^T is symmetric.

For any matrix A, AA^T and A^TA are symmetric.

Pivot variables

Pivot variables are the variables with leading elements attached to them when matrix is in REF. eg:

2x + 3y + z = 5

0x + y + 2z = 1 (this matrix has infinitely many solutions)

0x + 0y + 0z = 0

y & x are pivot variables.