Proof Theorem MMA Matrix Multiplication is Associative Suppose A A is an m×n m × n matrix, B B is an n×p n × p matrix and D D is a p×s p × s matrix. A. The first is that if the ones are relaxed to arbitrary reals, the resulting matrix will rescale whole rows or columns. That is if C,B and A are matrices with the correct dimensions, then (CB)A = C(BA). Proof We will concentrate on 2 × 2 matrices. Relevant Equations:: The two people that answered both say the order doesn't matter since matrix multiplication is associative: (A*A)*A=A*(A*A) But I actually don't get the same matrix. The main condition of matrix multiplication is that the number of columns of the 1st matrix must equal to the number of rows of the 2nd one. Proof: Since matrix-multiplication can be understood as a composition of functions, and since compositions of functions are associative, it follows that matrix-multiplication is associative Theorem 4 Given matrices A 2Rm n and B 2Rn p, the following holds: r(AB) = (rA)B = A(rB) Proof: First we prove r(AB) = (rA)B: r(AB) = r h Ab;1::: Ab;p i = h rAb;1::: rAb;p i Special types of matrices include square matrices, diagonal matrices, upper and lower triangular matrices, identity matrices, and zero matrices. However, this proof can be extended to matrices of any size. In standard truth-functional propositional logic, association, or associativity are two valid rules of replacement. The Associative Property of Multiplication of Matrices states: Let A , B and C be n × n matrices. L ( R m, R n) → R n × m. so that every T ∈ L ( R m, R n) is associated with a unique matrix M T ∈ R n × m. It turns out that this correspondence is particularly nice, because it satisfies the following property: for any T ∈ L ( R m, R n) and any S ∈ L ( R n, R k), we have that. Here it is for the 1st row and 2nd column: (1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12 = 64 We can do the same thing for the 2nd row and 1st column: (4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11 = 139 And for the 2nd row and 2nd column: (4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12 = 154 And w… Let be a matrix. Proof Proposition (associative property) Matrix addition is associative, that is, for any matrices, and such that the above additions are meaningfully defined. Let , , be any arbitrary 2 × 2 matrices with real number entries; that is, = μ ¶ = μ ¶ = μ ¶ where are real numbers. M S M T = M S ∘ T. So you have those equations: Example 1: Verify the associative property of matrix multiplication for the following matrices. Then, (AB)C = A(BC) . 3. A matrix is usually denoted by a capital letter and its elements by small letters : a ij = entry in the ith row and jth column of A. 1. For any matrix A, ( AT)T = A. Then (AB)C = A(BC). The proof of Theorem 2. B. 2. Use the multiplicative property of determinants (Theorem 1) to give a one line proof that if A is invertible, then detA 6= 0. Let the entries of the matrices be denoted by a11, a12, a21, a22 for A, etc. Matrix-Chain Multiplication • Let A be an n by m matrix, let B be an m by p matrix, then C = AB is an n by p matrix. So you get four equations: You might note that (I) is the same as (IV). Then A(BD) =(AB)D A (B D) = (A B) D. Theorem 7 If A and B are n×n matrices such that BA = I n (the identity matrix), then B and A are invertible, and B = A−1. But first, a simple, but crucial, fact about the identity matrix. Basically all the properties enjoyed by multiplication of real numbers are inherited by multiplication of a matrix by a scalar. 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