COS 302 Precept 1 - Princeton University
COS 302 Precept 1
Princeton University
Spring 2020
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COS 302 Precept 1
Spring 2020 1 / 34
Table of Contents
1 Properties of Matrices 2 Brute Force 3 Row-Echelon Form 4 Reduced Row-Echelon Form 5 Elementary Transformations 6 Gaussian Elimination
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Table of Contents
1 Properties of Matrices 2 Brute Force 3 Row-Echelon Form 4 Reduced Row-Echelon Form 5 Elementary Transformations 6 Gaussian Elimination
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Matrix Addition
Let X represent a matrix, Xij denote the entry that is in the ith row and jth column of X . (A + B)ij = Aij + Bij
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Matrix Multiplication
Let A Rm?n, B Rn?k
(AB)ij =
n l =1
Ail
Blj
In general, matrix multiplication is not commutative.
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Properties of Matrix Multiplication
Associativity: A Rm?n, B Rn?p, C Rp?q : (AB)C = A(BC )
Distributivity: A, B Rm?n, C , D Rn?p, (A + B)C = AC + BC , A(C + D) = AC + AD
Multiplication By Identity Matrix: A Rm?n, ImA = AIn = A, where Im is an m ? m matrix such that it has 1s on the diagonal and 0s everywhere else. It is known as the identity matrix.
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Transpose
ATij = Aji If AT = A, A is known as a symmetric matrix.
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Matrix Transpose Properties
(AT )T = A (A + B)T = AT + BT (AB)T = BT AT
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