Linear Algebra MATH 13

Elementary row operations

1. Interchange two rows 2. Multiply a row by a nonzero constant 3. Add a multiple of a row to another row

Matrix in row-echelon form properties

1. Rows of only zeros are at the bottom 2. For each row that's not entirely zeros, the first nonzero entry is 1 (leading 1) 3. For two successive nonzero rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row

Determinant of an invertible matrix thm

A square matrix A is invertible (nonsingular) if and only if det(A)≠0

A property of invertible matrices thm

A square matrix A is invetible if and only if it can be written as the product of elementary matrices

Gauss-Jordan elimination

Algorithm to reduce a matrix to its reduced row-echelon form

Elementary Matrix def

An n x n matrix that can be obtained from the identity matrix In by a single row operation

Meaning of solving an equation

Finding the solution set

Transpose of a matrix

Formed by writing the columns of the matrix as rows

Definition of Matrix Addition

If A = [aij] and B = [bij] are matrices of size m x n, then their sum is the m x n matrix given by A + B = [aij + bij]. (sum of two matrices of different sizes is undefined)

Definition of Matrix Multiplication

If A = [aij] is an m x n matrix and B = [bij] is and n x p matrix, then the product AB is an m x p matrix AB = [cij] where cij = ∑aik bkj =ai1b1j + ai2b2j + ai3b3j +...+ain bnj (to find the entry in the ith row and the jth column of the product AB, multiply the entries in the ith row of A by the corresponding entries in the jth column of B and then add the results)

The inverse of a product thm

If A and B are invertible matrices of order n, then AB is invertible and (AB)^-1 = B^-1 A^-1

Properties of transposes Thm

If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true 1. (A^t)^t = A 2. (A + B)^t = A^t + B^t 3. (cA)^t = c(A^t) 4. (AB)^t = B^tA^t

Determinant of a Matrix Product thm

If A and B are square matrices of ordern n, then det(AB) = det(A) det(B)

Properties of the identity matrix Thm

If A is a matrix of size m x n, then the following properties are true. 1. AIn = A 2.ImA = A

Conditions that yield a zero determinant thm

If A is a square matrix and any one of the following conditions is true, then det(A) = 0 1. An entire row ( or col ) consists of zeros 2. Two rows (or columns) are equal 3. One row (or col ) is a multiple of another row ( or col )

Determinant of a Scalar Multiple of a Matrix thm

If A is a square matrix of order n and c is a scalar, then the determinant of cA ids det(cA) = cⁿ det(A)

Determinant of a transpose thm

If A is a square matrix, then det(A) = det(A^t)

Determinant of a triangular matrix thm

If A is a triangular magtrix of order n, then its determinant is the product of the enttries on the main diagonal. That is, |A| = a11a22a33...ann

Uniqueness of an Inverse Matrix Thm

If A is an invertible matrix, then its inverse is unique. The inverse of A is denoted by A^-1

Systems of equation with unique solutions thm

If A is an invertible matrix, then the system of linear equations Ax = b has a unique solution given by x = A^-1 b

Properties of zero matrices Thm

If A is an m x n matrix and c is a scalar, then the following properties are true. 1. A + Omn = A 2. A + (-A) = Omn 3. If cA = Omn, then c = 0 or A = Omn

Determinant of an inverse matrix thm

If A is an n x n invertible matrix, then det(A^-1) = 1/det(A)

Equivalent conditions for a Nonsingular Matrix thm

If A is an n x n matrix, then the following statements are equivalent. 1. A is invertible 2. Ax = b has a unique solution for every n x 1 column matrix b 3. Ax = 0 has only the trivial solution 4. A is row-equivalent to In 5. A can be written as the product of elementary matrices 6. det(A) ≠ 0

Properties of matrix multiplication Thm

If A, B and C are matrices (with sizes such that the given matrix products are defined), and c is a scalar, then the following properties are true. 1. A(BC) = (AB)C associative property of multiplication 2. A(B + C) = AB + AC distributive property 3. (A + B)C = AC + BC distributive property 4. c(AB) = (cA)B = A(cB)

Cancellation properties thm

If C is an invertible matrix, then the following properties hold. 1. if AC = BC, then A = B (right cancellation ppty) 2. if CA = CB, then A = B (left cancellation ppty)

Elementary Matrices are invertible thm

If E is an elementary matrix, then E^-1 exists and is an elementary matrix

Properties of inverse Matrices

If is an invertible matrix, k is a positive integer, and c is a nonzero scalar, then A^-1, A^k, cA and A^t are invertible and the following are true: 1. (A^-1)^-1 = A 2. (A^k)^-1 = A^-1 A^-1....A^-1 = (A^-1)^k 3. (cA)^-1 = 1/c A^-1 4. (A^t)^-1 = (A^-1)t

LU-factorization

If the n x n matrix A can be written as the product of a lower triangular matrix L and an upper triangular matrix U, then A = LU is an LU-factorization of A

Matrix in reduced row-echelon form properties

It's in row-echelon form and every column that has a leading one has zeros in every position above and below its leading 1

Elementary row operations and Determinants thm

Let A and B be square matrices. 1. When B is obtained from A by interchanging two rows of A, det(B) = -det(A) 2. When B is obtained from A by adding a multiple of a row of A to anbother row of A, det(B) = det(A) 3. When B is obtained from A by multiplyingh a row of A by a nonzero constant c, det(B) = c det(A)

Finding inverse of a matrix by gauss-jordan elimination

Let A be a square matrix of order n 1. Write the n x 2n matrix that consists of the given matrix A on the left and the n x n identity matrix I on the right to obtain [A | I] (adjoining matrix I to matrix A) 2. If possible, row reduce A to I. The result will be the matrix [I | A^-1]. If this is not possible then A is singular 3. Check by multiplying to see that AA^-1 = I = A^-1 A

Expansion by cofactors thm

Let A be a square matrix of order n. Then the determinant of A is given by det(A) =∑ aijCij = ai1Ci1 + ... + ainCin (ith row exp.) or det(A) =∑ aijCij = a1jC1j + ... + anjCnj (jth column exp)

Representing elementary row operations thm

Let E be the elementary matrix obtained by performing an elementary row operation on Im. If that same elementary row operation is performed on an m x n matrix A, the the resulting matrix is given by the product EA

Symmetric matrix

Matrix for which A = A^T . if A = [aij], then aij = aji for all i ≠ j

Consistent system

System of linear equations that has at least one solution

Inconsistent system

System that has no solution

Homogeneous system

Systems of linear equations in which each of the constant terms is zero. It always has at least one solution (trivial solution)

Equivalent systems of linear equations

Systems that have the same solution set

Definition of Equality of Matrices

Two matrices A = [aij] and B = [bij] are equal when they have same size (m x n) and aij = bij for 1≤ j ≤n

Definition of the inverse of a matrix

an n x n matrix A is invertible (or nonsingular) when there exists an n x n matrix B such that AB = BA = In where In is the identity matrix of order n. The matrix B is called the (multiplicative) inverse of A. A matrix that does not have an inverse is called noninvertible (or singular)

Definition of Scalar Multiplication

if A = [aij] is an m x n matrix and c is a scalar, then the scalar multiple of A by c is the m x n matrix given by cA = [caij]

Properties of matrix addition and scalar multiplication Thm

if A,B and C are m x n matrices, and c and d are scalars, then the following properties are true. 1. A + B = B + A commutative property of addition 2. A + (B + C) = (A + B) + C associative property of addition 3. (cd)A = c(dA) associative property of multiplication 4. 1A = A multiplicative identity 5. c(A + B) = cA + cB distributive property I 6. (c + d)A = cA + dA distributive property II

Solution set

set of all solutions of a linear equation. It's described entirely by a parametric representation