02 Matrix Algebra Flashcards
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 ______. That is,
AB = ______
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 b2 … bp] = [Ab1 Ab2 … Abp]
Each column of AB is a __ of the columns of A using ____ from the corresponding column of B
Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B
rowi(AB) = __
rowi(AB) = rowi(A) B
The cancellation laws do ___ for matrix multiplication. That is, if AB = AC, then it is ___ in general that B = C.
The cancellation laws do not hold for matrix multiplication. That is, if AB = AC, then it is not true in general that B = C.
If a product AB is the zero matrix, you ___ conclude in general that either A = 0 or B = 0.
If a product AB is the zero matrix, you cannot conclude in general that either A = 0 or B = 0.
(A + B)T =
(A + B)T = AT + BT
(AB)T =
(AB)T = BTAT
The transpose of a product of matrices equals the product of their transposes _____
The transpose of a product of matrices equals the product of their transposes in the reverse order.
I (identity matrix) =
I = A-1A = AA-1
If ____, then A is invertible and
A-1 = ___
If ad - bc != 0, then A is invertible and

If A is an invertible n x n matrix, then for each b in Rn, the equation Ax = b has the ____ solution _____.
If A is an invertible n x n matrix, then for each b in Rn, the equation Ax = b has the unique solution x = A-1b.
If A and B are n x n invertible matrices, then ____
If A and B are n x n invertible matrices, then so is AB
(AB)-1 =
(AB)-1 = B-1A-1
(AT)-1 =
(AT)-1 = (A-1)T
The product of n x n invertible matrices is invertible, and the inverse is the product of their inverses ___
The product of n x n invertible matrices is invertible, and the inverse is the product of their inverses in the reverse order
An elementary matrix is one that is obtained by ____.
An elementary matrix is one 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 ____.
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 Im.
____ is the elementary matrix of the same type that transforms E back into I.
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 A is invertible if and only if A is row equivalent to __, and in this case, any sequence of elementary row operations that reduces A to __ also transforms ___.
An n x n matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms Ininto A-1.
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.
The Invertible Matrix Theorem
Let A be square n x n matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false.
- A is an invertible matrix.
- A is row equivalent to the n x n identity matrix.
- A has n pivot positions.
- The equation Ax = 0 has oinly the trivial solution.
- The columns of A form a linearly independent set.
- The linear transformation x |-> Ax is one-to-one.
- The equation Ax = b has at least one solution for each b in Rn.
- The columns of A span Rn.
- The linear transformation x|-> Ax maps Rn onto Rn.
- There is an n x n matrix C such that CA = I.
- There is an n x n matrix D such that AD = I.
- AT is an invertible matrix.
Let A and B be square matrices. If AB = I, then A and B are ____
Let A and B be square matrices. If AB = I, then A and B are both invertible, with B = A-1 and A = B-1.
invertible =
noninvertible =
invertible = nonsingular
noninvertible = singular
If the columns of a 4 x 3 matrix are linearly independent, ____ solutions to Ax = b.
If the columns of a 4 x 3 matrix are linearly independent, we cannot use the IMT to conclude anything about the existence or nonexistence of solutions to equations of the form Ax = b.