Properties of Matrix Operations
0011 0010 1010 1101 0001 0100 1011
Mark Ginn Math 2240 Appalachian State University
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Properties of Matrix Addition
0011 0010 1010 1101 0001 0100 1011
A+B = B+A A+(B+C)=(A+B)+C (cd)A = c(dA) 1A=A c(A+B) = cA+cB (c+d)A = cA +dA A+0mn=A A+(-A) = 0mn If cA=0mn then c=0 or A=0mn.
Commutative Associative Scalar Associative Scalar identity Scalar distributive 1 Scalar distributive 2 Additive identity Additive Inverse Scalar cancellation property
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0011 0010 1010 1101 0001 0100 1011
Properties of Matrix Multiplication
– – – – – – Associative Left distributive Right Distributive Scalar Associative Multiplicative Identity Multiplicative Identity
A(BC) = (AB)C A(B+C) = AB +AC (A+B)C = AC+BC c(AB) = (cA)B=A(cB) AIn = A ImA = A assuming A is m by n and all operations are defined.
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Using Properties to Prove Theorems
0011 0010 1010 1101 0001 0100 1011
• Using these properties we can prove the following theorem (which we have already been assuming).
• Theorem: For a system of linear equations in n variables, precisely one of the following is true:
1. The system has exactly one solution. 2. The system has an infinite number of solutions. 3. The system has no solutions.
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The Transpose of a Matrix
0011 0010 1010 1101 0001 0100 1011
• We will find it useful at times to talk about the transpose of a matrix. • Given an m by n matrix A, we define AT (A transpose) to be the n by m matrix:
a1,1 a1,2 T A a1,n a2,1 am ,1 a2,2 am,2 . a2,n am,n
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Properties of Transposes
0011 0010 1010 1101 0001 0100 1011
1. (AT)T = A
2. (A + B) T = AT+BT 3. (cA)T = c(AT)
Transpose of a transpose
Transpose of a sum
4. (AB)T = BTAT
Transpose of a scalar product Transpose of a product
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What about Mult. Inverses
0011 0010 1010 1101 0001 0100 1011
• For an n by n matrix A, can we find an n by n matrix A-1 so that AA-1=A-1A=In ?
• Does this always work? • Hmwk: p. 62: 4,5,10,14,16,28,30,34,42,51 p. 75: 5,14,18,28,33
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