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