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22 Cards in this Set
- Front
- Back
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DIAGONAL MATRIX
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A square matrix whose nondiagonal entries are 0.
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ZERO MATRIX
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An m x n matrix whose entries are all 0, written as 0.
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Two matrices are equal iff…
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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.
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SUM
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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.
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SCALAR MULTIPLE
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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.
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ALGEBRAIC PROPERTIES OF SCALAR AND ADDITIVE MATRICES
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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
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MATRIX MULTIPLICATION
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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]
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What can be said about the columns of the product matrix AB?
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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.
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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?
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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)
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Does AB = BA?
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Not necessarily. Position is critical in matrix multiplication. If AB = BA, then A and B commute with each other.
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Do cancellation laws hold for matrix multiplication?
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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.
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POWERS OF A MATRIX
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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.
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TRANSPOSE
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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.
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What are the characteristics of matrix transposes?
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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.
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INVERTIBLE MATRIX
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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.
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If A in an invertible n x n matrix, then for each bεRn, the equation Ax = b…
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has the unique solution x = A(-1)b.
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PROPERTIES OF INVERTIBLE MATRICES
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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
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ELEMENTARY MATRIX
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A matrix that is obtained by performing a single elementary row operation on an identity matrix.
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If an elementary row operation is performed on an m x n matrix A, the resulting matrix can be written as…
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EA, where the m x m matrix E is created by performing the same row operation on I(m).
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Each elementary matrix E is invertible. The inverse of E is…
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the elementary matrix of the same type that transforms E back into I.
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An n x n matrix is invertible iff…
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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).
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What is the algorithm for finding A(-1)?
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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.
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