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![LINEAR ALGEBRA
Properties of Identity Matrices
𝑰 𝟏 =
For a positive integer, 𝑰 𝒏 would denote the square matrix of
order n whose main diagonal (left to right) entries are 1 and
the rest of the entries are zero (Identity Matrix).
𝟏 𝟎
𝟎 𝟏
𝑰 𝟑 =
𝟏 𝟎 𝟎
𝟎 𝟏 𝟎
𝟎 𝟎 𝟏
[𝟏] 𝑰 𝟐 =](https://image.slidesharecdn.com/finalreportinlinearalgebra-180311062317/85/Algebraic-Properties-of-Matrix-Operations-11-320.jpg)
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Algebraic Properties of Matrix Operations
- 1. HELLO! ARE YOU READY? IF YES, LET’S START.
- 2. LINEAR ALGEBRA Solving aMatrix Equation a) 3X + A = B A = Using your skills in matrix addition and scalar multiplication, solve the following matrix equation: 𝟏 −𝟐 𝟎 𝟑 B = −𝟑 𝟒 𝟐 𝟏 b) 2X + 2A = 4B
- 3. M117 Linear Algebra Lesson 4 AlgebraicProperties of Matrix Operations REPORTERS: Joycee Anne Pura Kristopher Hiloma BSED ─ MATHEMATICS
- 4. LINEAR ALGEBRA Let A,B and C be m x n matrices. Properties of Matrix Addition A +B = B +A A + (B +C ) = (A +B ) + C A = 𝟒 ￚ𝟏 𝟐 𝟑 B = 𝟏 ￚ𝟐 𝟓 𝟕 C = ￚ𝟓 𝟑 ￚ𝟕 ￚ𝟏𝟎 Commutativity of Addition Associativity of Addition A +B or B +A = 𝟓 ￚ𝟑 𝟕 𝟏𝟎 B +C = ￚ𝟒 𝟏 ￚ𝟐 ￚ𝟑 𝟎 𝟎 𝟎 𝟎 Zero Matrix
- 5. LINEAR ALGEBRA Let Aand B be m x n matrices and c and d are scalars. Properties of Scalar Multiplication (cd) A = c (dA) 1A = A c (A + B ) = cA + cB (c + d) A = cA + dA Associative Property of Multiplication Multiplicative Identity Distributive Property Distributive Property A = 𝟒 ￚ𝟏 𝟐 𝟑 B = 𝟏 ￚ𝟐 𝟓 𝟕 c = 2 d = 𝟏 𝟐
- 6. LINEAR ALGEBRA Properties ofZero Matrices A + 𝑶 𝒎𝒏 = A A = 𝟒 ￚ𝟐 ￚ𝟓 𝟐 𝟑 ￚ𝟏 𝑶 𝟐𝟑 = 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 The m x n matrix with all entries of zero is denoted by 𝑶 𝒎𝒏 , for a matrix A of size m x n, we have: This property states that 𝑶 𝒎𝒏 or the zero matrix is the additive identity for the set of all m x n matrices.
- 7. LINEAR ALGEBRA Properties ofZero Matrices A + D = 𝑶 𝒎𝒏 A = This property states that ￚ A or the negative of A is the additive inverse of A . A + (ￚ A ) = 𝑶 𝒎𝒏 𝟒 ￚ𝟐 ￚ𝟓 𝟐 𝟑 ￚ𝟏 ￚ A = ￚ𝟒 𝟐 𝟓 ￚ𝟐 ￚ𝟑 𝟏 ￚ A = ￚ1(A) The m x n matrix with all entries of zero is denoted by 𝑶 𝒎𝒏 , for a matrix A of size m x n, we have:
- 8. LINEAR ALGEBRA Properties ofZero Matrices If cA = 𝑶 𝒎𝒏 , then c = 0 or A = 𝑶 𝒎𝒏 . A = 𝟒 ￚ𝟐 ￚ𝟓 𝟐 𝟑 ￚ𝟏 cA = A = 𝑶 𝒎𝒏 0 𝟒 ￚ𝟐 ￚ𝟓 𝟐 𝟑 ￚ𝟏 cA = 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 c = 𝟏𝟎 cA = 10 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 cA = 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 c = 0 The m x n matrix with all entries of zero is denoted by 𝑶 𝒎𝒏 , for a matrix A of size m x n and a scalar c, we have:
- 9. LINEAR ALGEBRA What didyou notice? For these first three properties of matrix operations, “matrices behave like real numbers.” Properties of Matrix Addition Properties of Zero Matrices Commutativity of Addition Associativity of Addition Additive Identity Additive Inverse Zero (0) as a Multiplier Properties of Scalar Multiplication
- 10. LINEAR ALGEBRA Let A,B and C be matrices with sizes such that the given matrix products are defined and c be a scalar. Properties of Matrix Multiplication a) A (BC) = (AB) C b) (A + B) C = AC + BC c) C (A + B) = CA + CB d) c (AB) = (cA)B + A(cB) Associativity Matrix Product Right Distributive Property Left Distributive Property
- 11. LINEAR ALGEBRA Properties ofIdentity Matrices 𝑰 𝟏 = For a positive integer, 𝑰 𝒏 would denote the square matrix of order n whose main diagonal (left to right) entries are 1 and the rest of the entries are zero (Identity Matrix). 𝟏 𝟎 𝟎 𝟏 𝑰 𝟑 = 𝟏 𝟎 𝟎 𝟎 𝟏 𝟎 𝟎 𝟎 𝟏 [𝟏] 𝑰 𝟐 =
- 12. LINEAR ALGEBRA Properties ofIdentity Matrices a) A 𝑰 𝒏 = A A = 𝑰 𝒏 = b) 𝑰 𝒎A = A A = For a positive integer, 𝑰 𝒏 and 𝑰 𝒎 would denote the square matrices of order n and m, respectively, whose main diagonal (left to right) entries are 1 and the rest of the entries are zero (Identity Matrix). 𝟑 −𝟐 𝟒 𝟎 −𝟏 𝟏 𝟏 𝟎 𝟎 𝟏 𝑰 𝒎 = 𝟏 𝟎 𝟎 𝟎 𝟏 𝟎 𝟎 𝟎 𝟏 −𝟐 𝟏 𝟒 Multiplicative Identity
- 13. LINEAR ALGEBRA Let A,B and C be matrices with sizes such that the given matrix products are defined and c be a scalar. Properties of Transposes a) (𝑨 𝑻 ) 𝑻 = A b) (A + B ) 𝑻 = 𝑨 𝑻 + 𝑩 𝑻 c) (cA) 𝑻 = c(𝑨 𝑻 ) d) (AB ) 𝑻 = 𝑩 𝑻 𝑨 𝑻 Transpose of a transpose Transpose of a sum Transpose of a scalar multiple Transpose of a product
- 14.