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Multiplication by scalars: if A is a matrix of size m × n and c is a scalar, then cA is a matrix of size m × n Matrix multiplication: if A is a matrix of size m × n and B is a matrix of size n × p, then the product AB is a matrix of size m × p

Inverting Matrices P Danziger Matrix Algebra Theorem 3 (Algebraic Properties of Matrix Multiplication) 1 (k + l)A = kA + lA (Distributivity of scalar multiplication

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from the fact that the sum of two matrices is only defined when they have the same size Properties of matrix multiplication (MM1): The product (AB)C is defined

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18 fév 2002 · matrix addition and scalar multiplication, is a vector space 2 Properties of Matrix Multiplication and In- verse Matrices Theorem 5 Let A, B and

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2 nov 2005 · The number of columns of A must be equal to the number of rows of B If A is 2 × n and B is n × 1 then AB is 2 × 1 This rule for multiplication may be extended to matrices, A, which have more than two rows

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B Properties of Matrix Multiplication: Theorem 1 2 Let A, B, and C be matrices of appropriate sizes Then the following properties hold: a) A(BC)=(AB)C

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1 fév 2012 · Definition A square matrix A is symmetric if AT = A Properties of transpose (1) ( AT )T = A (2) (A + B)T =

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Scalar multiplication: to multiply a matrix A by a scalar r, one That is, matrices are multiplied row by column: ( ∗ ∗ ∗ Properties of matrix multiplication:

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matrix multiplication, matrix-vector product • matrix Properties of matrix addition we can multiply a number (a k a scalar) by a matrix by multiplying every

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(A + B)C = AC + BC and A(B + C) = AB + AC Note: Matrix-matrix multiplication does not commute Only in very rare cases does AB equal BA Indeed, the

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Properties of matrix operations

The operations are as follows:

Addition: ifAandBare matrices of the same sizemn, thenA+B, their sum, is a matrix of sizemn. Multiplication by scalars: ifAis a matrix of sizemnandcis a scalar, thencAis a matrix of sizemn. Matrix multiplication: ifAis a matrix of sizemnandBis a matrix of sizenp, then the productABis a matrix of sizemp. Vectors: a vector of lengthncan be treated as a matrix of sizen

1, and the operations of vector addition, multiplication by scalars, and

multiplying a matrix by a vector agree with the corresponding matrix operations. Transpose: ifAis a matrix of sizemn, then its transposeATis a matrix of sizenm. Identity matrix:Inis thennidentity matrix; its diagonal elements are equal to 1 and its odiagonal elements are equal to 0. Zero matrix: we denote by 0 the matrix of all zeroes (of relevant size). Inverse: ifAis asquarematrix, then its inverseA1is a matrix of the same size. Not every square matrix has an inverse! (The matrices that have inverses are calledinvertible.) The properties of these operations are (assuming thatr;sare scalars and the sizes of the matricesA;B;Care chosen so that each operation is well dened):

A+B=B+A;(1)

(A+B) +C=A+ (B+C);(2)

A+ 0 =A;(3)

r(A+B) =rA+rB;(4) (r+s)A=rA+sA;(5) r(sA) = (rs)A; (6)

A(BC) = (AB)C;(7)

A(B+C) =AB+AC;(8)

(B+C)A=BA+CA;(9) 1 r(AB) = (rA)B=A(rB);(10) I mA=A=AIn; (11) (AT)T=A;(12) (A+B)T=AT+BT;(13) (rA)T=rAT;(14) (AB)T=BTAT;(15) (In)T=In; (16) AA

1=A1A=In;(17)

(rA)1=r1A1; r6= 0;(18) (AB)1=B1A1;(19) (In)1=In;(20) (AT)1= (A1)T;(21) (A1)1=A:(22) We see that in many cases, we can treat addition and multiplication of matrices as addition and multiplication of numbers. However, here are some dierences between operations with matrices and operations with numbers: Note the reverse order of multiplication in (15) and (19). (19) can only be applied if we know that bothAandBare invertible. In general,AB6=BA, even ifAandBare both square. IfAB=BA, then we say thatAandBcommute. For a general matrixA, we cannot say thatAB=ACyieldsB=C. (However, if we know thatAis invertible, then we can multiply both sides of the equationAB=ACto the left byA1and getB=C.) The equationAB= 0 does not necessarily yieldA= 0 orB= 0. For example, take A=1 0 0 0 ; B=0 0 0 1 2quotesdbs_dbs20.pdfusesText_26

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