Linear algebra

What is a Consistent system

a system with at least one solution

What is a inconsistent system

system with no solution

What does a unique solution look like graphically?

lines intersect

What does a infinite solution look like graphically?

lines are stacked or the same

What does no solution look like graphically?

lines are parallel

What is row echelon form?

1. In triangular form
2. If there are rows whose entries are all 0, they are below the rows having nonzero entries

What is Reduced Row echelon form?

1. In row echelon form
2. The first nonzero entry in each row is the only nonzero entry in its column

Guass - Jordan Elimination

Using row elementary operations to solve the system to reduced row echelon form

Scalar

the entries of a matrix, or real numbers.

Transpose

switch rows with columns and columns with rows

Symmetric

if transpose is the same as original matrix
A^T = A

Skew symmetric

A^T = -A

Linear system

Ax=b where A is an m x n matrix, x is an unknown vector in R^n (R is all real numbers), and b is in R^m. We consider first the case of one equation in several unknowns.

Consistency theorem

A linear system Ax=b is consistent if and only if b can be written as a linear combination of the column vectors of A.

How do you find the inverse of a matrix?

A^-1 = (adj A) / (det A)
- where A is a square matrix
- det(A) does not = 0

determinant of 2x2 matrix

det(A) = ad - bc

determinant of 3x3 matrix

a(ei - fh) - b(di - fg) + c(dh - eg)

adjoint matrix - adj(A)

the transpose of the cofactor matrix

Singular matrix

A singular matrix is a square matrix with no inverse. It's determinant is zero.

Nonsingular matrix

a matrix that is invertible and determinant is not 0

Transpose Property: (A^T)^T = ?

A

Transpose Property: (A+B)^T = ?

A^T + B^T

Transpose Property: (cA)^T = ?

cA^T

Transpose Property: (AB)^T = ?

B^T * A^T

Transpose Property: det(A^T)

det(A)

Transpose Property: (A^T)^-1 = ?

(A^-1)^T

Inverse property: (A^-1)^-1 = ?

A

Inverse Property: (AB)^-1 = ?

B^-1A^-1

Inverse Property: (ABC)^-1 = ?

C^-1B^-1A^-1

Inverse Property: (kA)^-1 = ?

(1/k)A^-1

When does AB = I_n ?

when A and B are inverses of each other

Determinant Properties: If I_n is the identity of order n x n, then det(I) = ?

1

If the matrix M^T is the transpose of matrix M< then det(M^T) = ?

det(M)

if Matrix M^-1 is the inverse of matrix M, then det(M^-1) = det(M)^-1 = ?

1/det(M)

if two square matrices M and N have the same size, then det(MN) = ?

det(M)det(N)

If matrix M has a size a x a and C is a constant, then det(CM) =?

C^adet(M)

How do you solve inverse of 2×2 Matrix A?

adj A/ det|A| where adjA =
|d -b|
|-c a|

How do you solve inverse of 3×3 Matrix A?

Adjoin matrix A with a Identity matrix on the right and use elementary row operations until the A becomes I.

How do you find all 3×3 matrices X given A and B in AX = B?

1. AX=B

2. A^-1AX = A^-1B

3. IX = A^-1B

4. X = A^-1B

Nilpotent matrix

Matrix multiplied to itself is a null matrix or 0

Cramer’s Rule

Swap answer column with first column in matrix A to get x, swap answer column with next column in A to get y and so on.
x = det(A_x)/ det(A)
y = det(A_y)/ det(A)

How do you solve LU Factorization?

Identity matrix to the left of the augmented matrix, use row operations until left side is a lower triangle and the identity matrix has become an upper triangle.

How do you solve for all solutions to Ax=b given using LU factorization?

Solve Ly = b by back substitution to get y then solve Ux = y with back substitution to get x values.