Chapter 1 Definitions Flashcards
What is a matrix?
A matrix is a rectangular array of objects
What is a reduced matrix?
A matrix is said to be reduced if each of the following conditions holds:
(i) the first nonzero entry in each row is 1, and in the column where this leading 1 occurs, all other entries are zero.
(ii) the leading one in each row lies to the right of the leading one in the preceding row
(iii) all rows consisting entirely of zeros lies at the bottom of the matrix
What are equal matrices?
Two matrices are equal if and only if they contain precisely the same entries
Row equivalent?
When one of the elementary row operations is used, we say the resulting matrix is row equivalent. Denoted by ~
What is a homogeneous system?
All constants(c) are 0
Theorem 1
Every homogeneous system is consistent
Theorem 2
A homogeneous system has an infinite number of solutions if n>m (unknowns>equations)
What is a square matrix?
where the number of rows equals the number of columns
What is a main diagonal?
(read L-R) the diagonal of a square matrix consists of the following entries (a[11], a[22],…,a[nm].
What is an Identity Matrix?
An identity matrix of order n is a matrix with 1’s on the main diagonal and zeros elsewhere, denoted by I[n]
Communative
A and B are called communative (AB=BA) if and only if AB=BA.
Zero Matrix
a matrix consisting entirely of zeros
Involutory
A is called involutory if and only if A^2 = I
Idempotent
A is called idempotent if and only if A^2 = A
Nilpotent
A called nilpotent if A^k = 0 matrix, for some positive integer k, the smallest such value of k is called index of nilpotency.
Diagonal Matrix
An n-square matrix is called diagonal if the entries above and below the diagonal are zeros.
Upper Triangular
An n-square matrix is called upper triangular if the entries below the main diagonal are zeros.
Lower Triangular
a lower triangular matrix consists of all zero entries above the main diagonal
Triangular
the matrix is upper and lower triangular
Theorem 3
i. A+0=A II.0A=0 III.Commutative Law for matrix addition IV.Associative law for matrix addition V.Distributive law for scalar multiplication VI. 1A = A VII.(a+b)A=aA+bA
Commutative Law for matrix addition
A + B = B + A
Associative Law for matrix addition
(A+B)+C= A + (B+C)
Distributive Law for Scalar Multiplication
a(A+B)=aA+aB
Nonsingular
If the inverse of A exists, A is called invertible, or nonsingular. A^-1
Theorem 4
Suppose A and B are nxn invertible matrices.
Then AB is invertible and (AB)^-1 = B^-1A^-1
Theorem 5
Suppose a sequence of matrices are all nxn invertible matrices. The inverse of the sequence is the inverses of all the matrices in reverse order
Theorem 6
Suppose A is invertible. Then A^-1 is also invertible, and (A^-1)^-1=A
Trace
Let A be an n-square matrix. The trace of A is the sum of the diagonal entries, denoted by tr(A).
Transpose
Let A be an mxn matrix. The transpose of A is the matrix obtained by making each row of A a column, while preserving the order, denoted by A^t
Theorem 7
Let A and B be mxn matrices.
(i) (A^t)^t = A
(ii) (A+B)^t = A^t + B^t
(iii) (AB)^t = B^tA^t
(iv) If A is invertible, then (A^-1)^t = (A^t)^-1
Symmetric
Let A be n-square matrix. A is called symmetric if A^t=A
Skew-Symmetric
A is called skew-symmetric if A^t=-A
Theorem 8
Let A and B be n-square symmetric matrices. AB is symmetric if and only if A and B commute
Elementary Matrix
An n-square matrix is called an elementary matrix if it can be obtained from the Identity matrix by means of a single elementary row operation.
Theorem 9
Every elementary matrix is invertible
Theorem 10
An n-square matrix is invertible if and only if it is the product of elementary matrices.
