Math 240 Sections 2.1-2.3, 2.6, 3.1-3.3, 4.1-4.3, 4.5 Exam Questions and Answers 100% Pass

Math 240 Sections 2.1-2.3, 2.6, 3.1-3.3, 4.1-4.3, 4.5 Exam Questions and Answers 100% Pass 2.1 Theorem 1 - Let A, B, and C be matrices of the same size and let r and s be scalars. a. A + B = B + A b. (A + B) + C = A + (B + C) c. A + 0 = A d. (r(A + B) = rA + rB e. (r + s)A = rA + sA f. r(sA) = (rs)A 2100% Pass Guarantee Katelyn Whitman All Rights Reserved © 2025 2.1 Definition - If A is an m x n matrix, and if B is an n x p matrix with columns b1 ... bp, then the product AB is the m x p matrix whose columns are Ab1...Abp. That is, AB = A[b1...bp] = [Ab1...Abp] Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B. 2.1 Theorem 2 - Let A be an m x n matrix and let B and C have sizes for which the indicated sums and products are defined. a. A(BC) = (AB)C b. A(B+C) = AB + AC c. (B + C)A = BA + CA d. r(AB) = (rA)B = (rB)A for any scalar r e. Im * A = A = A * In 2.1 Theorem 3 - Let A and B denote matrices whose sizes are appropriate for the following sums and products. a. (A^T)^T = A 3100% Pass Guarantee Katelyn Whitman All Rights Reserved © 2025 b. (A + B)^T = A^T + B^T c. For any scalar r, (rA)^T = rA^T d. (AB)^T = B^T * A^T The transpose of a product of matrices equals the product of their transposes in reverse order. 2.2 Theorem 5 - If A is an invertible n x n matrix then for each b in R^n, the equation Ax = b has the unique solution x = A^-1 * b 2.2 Theorem 6 - a. If A is an invertible matrix, then A^-1 is invertible and the inverse of it is A. b. If A and B are n x n matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. That is: