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### 22 Cards in this Set

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 DIAGONAL MATRIX A square matrix whose nondiagonal entries are 0. ZERO MATRIX An m x n matrix whose entries are all 0, written as 0. Two matrices are equal iff… they have th same size (the same number of rows and same number of columns) and if their corresponding columns are equal. This means that their corresponding entries are equal. SUM If A and B are m x n matrices, then the sum A + B is the m x n matrix whose columns are the sums of the corresponding columns in A and B. The sum A + B is defined only when A and B are the same size. SCALAR MULTIPLE If r is a scalar and A is a matrix, then the scalar multiple rA is the matrix whose columns are r times the corresponding columns in A. ALGEBRAIC PROPERTIES OF SCALAR AND ADDITIVE MATRICES 1. A + B = B + A 2. (A + B) + C = A + (B + C) 3. A + 0 = A 4. r(A + B) = rA + rB 5. (r + s)A = rA + sA 6. r(sA) = (rs)A MATRIX MULTIPLICATION If A is an m x n matrix, and if B is an n x p matrix with columb b1, …, bp, then the product AB is the m x p matrix whose columns are Ab1, …, Abp. AB = A[b1 b2 … bp] What can be said about the columns of the product matrix AB? Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B. The number of columns of A must therefore match the number of rows in B in order for a linear combination such as Ab1 to be defined. If A is an m x n matrix, and B and C have sizes for which sums and products are defined, what are the characteristics of multiplied matrices? 1. A(BC) = (AB)C 2. A(B + C) = AB + AC 3. (B + C)A = BA + CA 4. r(AB) = (rA)B = A(rB) for any scalar r 5. I(m)A = A = AI(m) Does AB = BA? Not necessarily. Position is critical in matrix multiplication. If AB = BA, then A and B commute with each other. Do cancellation laws hold for matrix multiplication? No. If AB = AC, then it is not true in general that B = C. Furthermore, if a product AB is the zero matrix, you cannot conclucde in gener that either A = 0 or B = 0. POWERS OF A MATRIX If A is an n x n matrix and if k I ap ositive integer, the A ^ k enotes the product of k copies of A. A ^ 0 is interpreted as the identity matrix. TRANSPOSE Fiven an m x n matrix A, the transpose of A is the n x m matrix, denoted by A^T, whose columns are formed from the corresponding rows of A. In other rows, make all of the columns rows and all the rows columns. What are the characteristics of matrix transposes? 1. (A ^ T) ^ T = A 2. (A + B) ^ T = A ^ T + B ^ T 3. For any scalar r, (rA) ^ T = rA ^ T 4. (AB) ^ T= (B ^ T)(A ^ T), or the transpose of a product of matrices equals the product of their transposes in the reverse order. INVERTIBLE MATRIX An n x n matrix A is said to be invertible if there is an n x n matrix C such that CA = I and AC = I. An invertible matrix is also called a nonsingular matrix. A matrix that is not invertible is also called a singular matrix. If A in an invertible n x n matrix, then for each bεRn, the equation Ax = b… has the unique solution x = A(-1)b. PROPERTIES OF INVERTIBLE MATRICES 1. If A is an invertible matrix, the A(-1) is invertible and [A(-1)](-1) = A 2. If A and B are n x n invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order, or (AB)(-1) = [B(-1)A(-1)] 3. If A is an invertible matrix, the so is A ^ T, and the inverse of A ^ T is the transpose of A(-1), or (A ^ T)(-1) = (A(-1)) ^ T ELEMENTARY MATRIX A matrix that is obtained by performing a single elementary row operation on an identity matrix. If an elementary row operation is performed on an m x n matrix A, the resulting matrix can be written as… EA, where the m x m matrix E is created by performing the same row operation on I(m). Each elementary matrix E is invertible. The inverse of E is… the elementary matrix of the same type that transforms E back into I. An n x n matrix is invertible iff… A is row equivalent to I(n), and in this case, any sequence of elementary row operations that reduces A to I(n) also transforms I(n) into A(-1). What is the algorithm for finding A(-1)? Row reduce the augmented matrix [A I]. If A is row equivalent to I, then [A I] is row equivalent to [I A(-1)]. Otherwise, A does not have an inverse.