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Questions and Answers
What is the largest number of pivot columns that a 4 × 5 matrix can have?
What is the largest number of pivot columns that a 4 × 5 matrix can have?
4
When A and B are n × n, invertible matrices, then AB is invertible and (AB)^{-1} = A^{-1}B^{-1}.
When A and B are n × n, invertible matrices, then AB is invertible and (AB)^{-1} = A^{-1}B^{-1}.
False
If A is an invertible n × n matrix, then the equation Ax = b is consistent for every b in R^n.
If A is an invertible n × n matrix, then the equation Ax = b is consistent for every b in R^n.
True
If A is an n × n matrix and Ax = e_j is consistent for every j, 1 ≤ j ≤ n, then A is invertible.
If A is an n × n matrix and Ax = e_j is consistent for every j, 1 ≤ j ≤ n, then A is invertible.
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There are some matrices A, B, C with AB = AC, A invertible and B ≠ C.
There are some matrices A, B, C with AB = AC, A invertible and B ≠ C.
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The equality A^T A = A A^T holds for all n × n matrices A.
The equality A^T A = A A^T holds for all n × n matrices A.
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If u and v are in R^n, how are uv^T and vu^T related?
If u and v are in R^n, how are uv^T and vu^T related?
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If an n × n matrix A has n pivot positions, then the reduced echelon form of A is the n × n identity matrix, I_n.
If an n × n matrix A has n pivot positions, then the reduced echelon form of A is the n × n identity matrix, I_n.
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What is L in the LU decomposition of an m × n matrix A?
What is L in the LU decomposition of an m × n matrix A?
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What is U in the LU decomposition of an m × n matrix A?
What is U in the LU decomposition of an m × n matrix A?
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What are the four fundamental subspaces?
What are the four fundamental subspaces?
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What does Column Space C(A) indicate?
What does Column Space C(A) indicate?
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What is Null Space N(A)?
What is Null Space N(A)?
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What is Row Space C(A^T)?
What is Row Space C(A^T)?
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What is the Left Null Space N(A^T)?
What is the Left Null Space N(A^T)?
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What does transpose mean?
What does transpose mean?
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How do you test for inverse of a matrix A?
How do you test for inverse of a matrix A?
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Study Notes
Pivot Columns in Matrices
- A 4 × 5 matrix can have at most 4 pivot columns, limited by the number of rows.
- Pivot columns contain leading non-zero entries in each row of its echelon form.
Invertible Matrices
- If A and B are invertible n × n matrices, the product AB is also invertible; however, (AB) ^−1 = B ^−1 A ^−1, not A ^−1 B ^−1.
Consistency of Equations with Invertible Matrices
- For any invertible n × n matrix A, the equation Ax = b is consistent for every vector b in R^n.
- If Ax = ej is consistent for every j (1 ≤ j ≤ n), then A must be invertible.
Relations Between Matrices
- If AB = AC and A is invertible, then B must equal C.
- The equality A TA = AAT does not hold for all n × n matrices A.
Matrix Transposition
- For vectors u and v in R^n, the matrices uvT and vuT are transposes of each other.
- Transposition properties: (AB) T = B TA T and (A T) T = A.
Identity Matrix in Reduced Echelon Form
- An n × n matrix A with n pivot positions has its reduced echelon form as the identity matrix I_n.
LU Decomposition
- In the LU decomposition of an m × n matrix A, L is an m × m lower triangular matrix with ones on the diagonal; U is the echelon form of A with pivots on the diagonal.
Four Fundamental Subspaces
- The four subspaces related to matrices are:
- Column Space (C(A))
- Null Space (N(A))
- Row Space (C(A^T))
- Left Null Space (N(A^T))
Column Space
- The column space C(A) is the span of the pivot columns of matrix A and is a subspace of R^m.
- If Ax = b is solvable, then vector b belongs to C(A).
Null Space
- The null space N(A) is defined as the set of all vectors x such that Ax = 0; it forms a subspace of R^n.
- N(A) = {0} when there is a pivot in each column.
Row Space
- The row space of a matrix A, denoted C(A^T), is the span of the rows of A and is a subspace of R^n.
Left Null Space
- The left null space N(A^T) consists of all solutions x such that x^T A = 0, forming a subspace of R^m.
Matrix Transpose Properties
- Transposition flips the matrix across its main diagonal, defined as (A^T)_ij = A_ji.
- Key properties include: (A^T)^T = A, (A+B)^T = A^T + B^T, and (AB)^T = B^TA^T.
Inverse Matrices
- An n × n matrix A has an inverse if it has a pivot in every row and column, leading to a unique solution for Ax = 0.
- The inverse is computed using the relationship A^-1 A = I and the property (AB) ^-1 = B^-1 A^-1.
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Test your knowledge on matrices with these flashcards. Each card includes a question and a definition to help you understand essential concepts. Perfect for students looking to strengthen their grasp of matrix theory.
