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Properties of Matrices

Properties of Matrices. Associative Property. Addition . Multiplication. ( ab )c = a( bc ) ( 1*2) * 3 = 1 * ( 2 * 3 ) The association – 2 with 1 or 2 with 3 – doesn’t matter. (a + b) + c = a + (b + c) (1 + 2) + 3 = 1 + (2 + 3 ) The association – 2 with 1 or 2 with 3 – doesn’t matter.

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Properties of Matrices

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  1. Properties of Matrices

  2. Associative Property Addition Multiplication (ab)c = a(bc) ( 1*2) * 3 = 1 * ( 2 * 3 ) The association – 2 with 1 or 2 with 3 – doesn’t matter • (a + b) + c = a + (b + c) • (1 + 2) + 3 = 1 + (2 + 3 ) • The association – 2 with 1 or 2 with 3 – doesn’t matter

  3. Associative Property • Will it work for matrices? • Consider these matrices… • What is (A + B) + C? • A + B = + =

  4. Associative Property • What is (A + B) + C? • A + B = + = • What is A +( B + C)? • B + C = • A + (B + C)= + =

  5. What does this mean? • Matrix Addition is Associative

  6. Associative Property • Will it work for multiplication of matrices? • Consider these matrices… • What is (A B)* C? • A B = • (AB)*C =

  7. Associative Property • What is A*( B C)? • B C= • A* (BC) =

  8. What does this mean? • Matrix Multiplication is Associative (at least if the matrices can be multiplied)

  9. Commutative Property Addition Multiplication ab = ba 1*2 = 2 * 1 The order doesn’t matter • a + b = b + a • 1 + 2 = 2 + 1 • The order doesn’t matter

  10. Commutative Property • Will it work for matrices? • Consider these matrices… • What is A+B? • A + B = • What is B+A? • B + A =

  11. What does this mean? • Matrix Addition is Commutative

  12. Commutative Property • Will it work for multiplication of matrices? • Consider these matrices… • What is AB? • A B = • What is BA? • B A =

  13. What does this mean? • Matrix Multiplication is NOT Commutative

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