Matrices And System Of Linear Equations

Front 1

For Matrix A what does mxn stand for?

Back 1

Rows x columns

Front 2

symmetric matrix

Back 2

A^T = A

Front 3

Skew-symmetric matrix

Back 3

A^T = -A

Front 4

Transpose

Back 4

The rows and columns switched to columns and rows

ex. : row1 becomes column1

Front 5

Matrix Vector Form

Back 5

Ax=b where x and b are vectors and A is a coefficient matrix

Front 6

E.R.O. Pij

Back 6

switching the ith and jth rows

Front 7

E.R.O. Mi(k)

Back 7

multiplication of ith row by scalar k

Front 8

E.R.O. Aij(k)

Back 8

adding the jth row to the k times the ith row

Front 9

Gaussian Elimimation

Back 9

Reducing matrix to row echelon form using elementary row operations then using the R.E.F. linear system to solve for all variables of system.

Front 10

Row Echelon Form

Back 10

Matrix where all rows made entirely of zeros is at the bottom; all other rows first non zero numer is one and is called a pivot; and the pivots of each row occur directly right of the above row's pivot.

Front 11

A matrix A is invertible if?

Back 11

the matrix A has n×n dimensions and there exists a n×n matrix B that makes AB=I=BA where I is the n×n identity matrix.

Front 12

Gauss-Jordan technique

Back 12

[A|I]~~E.R.O.~~[I|A^-1]

Front 13

Invertible Matrix Theorem

Back 13

The following is equavalent for A:
(a) A is invertible
(b) The equation Ax=b has a unique solution for every b in R^n
(c) The equation Ax=0 has only the trivial solutionx=0
(d) rank(A)=n
(e) A can be expressed as a product of elementary matrices
(f) A is row-equivalent to Indentity matrix

Front 14

Identity Matrix Properties

Back 14

AI=A
IA=A

Front 15

Transpose Properties

Back 15

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

Front 16

Matrix Multiplication Properties

Back 16

A(BC)=(AB)C
A(B+C)=AB+AC
(A+B)C=AC+BC