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aij
The scalar entry in the ith row and jth column (diagonal entries in an mxn matrix A are a11, a22, a33, … (i=j)
Mian Diagonal
Defined on an mxn matrix
Antidiagonal/Counter Diagonal
Defined only on a square matrix Anxn
aij, (i + j = n +1)
Diagonal Matrix
A square (n x n) matrix whose non diagonal entries are zero (ex = identity matrix In)
Zero Matrix
All entries are zero
When are two matrices equal?
If they have the same size and their corresponding entries are zero
Matrix Sums
If A and B are mxn matrices (same size), each entry in A + B is the sum of the corresponding entries
Scalar Multiples
If A is an mxn matrix and r is a scalar, then any entry of rA is r times the corresponding entry in A
Theorem 2.1
Let A, B, and C are matrices of the same size and let r and s be scalars
A + B = B + A
(A + B) + C = A + (B + C) = (A + C) +B
A + 0 = A
r(A + B) = rA + rB
(r + s)A = rA =sA
r(sA) = (rs)A = s(rA)
Matrix Multiplication
Bx = x1b1 + x1b1 + x2b2 + … + xnbn
If A is an mxn matrix, B is an nxp matrix with columns [b1 b2 … bp], then AB is the m x p matrix whose columns are Ab1, …, Abp
Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B
Row-Column Rule for Computing AB
If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding entries from row i of A and column j of B
(AB)ij = ai1b1j + ai2b2j + … + ainbnj
Properties of Matrix Multiplication: Theorem 2.2
Let A be an mxn matrix, and let B and C have sides for which the indicated sums and products are defined
A(BC) = (AB)C
A(B + C) = AB + AC]
(B +C)A = BA + CA
r(AB) = (rA)B = A(rB) for any scalars r
InA = A = AIn
Powers of a Matrix
If A is an nxn matrix, and if k is a positive integer, then
A^k = AAA…A (copies of A’s)
Transpose of a Matrix
Given an mxn matrix A, the transpose of A is the nxm matrix, denoted by A^T, whose column are formed from the corresponding dows of A (Aij = (A^T)ji)
Theorem 2.3
(A^T)^T = A
(A + B)^T = A^T + B^T
For any scalar r, (rA)^T = rA^T
(AB)^T = B^TA^T
