Matrices Flashcards

Questions and Answers

What is the largest number of pivot columns that a 4 × 5 matrix can have?

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.

True

If A is an n × n matrix and Ax = e_j is consistent for every j, 1 ≤ j ≤ n, then A is invertible.

True Signup and view all the answers

There are some matrices A, B, C with AB = AC, A invertible and B ≠ C.

False Signup and view all the answers

The equality A^T A = A A^T holds for all n × n matrices A.

False Signup and view all the answers

If u and v are in R^n, how are uv^T and vu^T related?

They are n × n transposes of one another. Signup and view all the answers

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.

True Signup and view all the answers

What is L in the LU decomposition of an m × n matrix A?

L is an m × m lower triangular matrix with ones on the diagonal. Signup and view all the answers

What is U in the LU decomposition of an m × n matrix A?

U is an m × n echelon form of A. Signup and view all the answers

What are the four fundamental subspaces?

Column Space, Null Space, Row Space, Left Null Space. Signup and view all the answers

What does Column Space C(A) indicate?

Ax = b is solvable if and only if b is in C(A). Signup and view all the answers

What is Null Space N(A)?

The set of all vectors when multiplied by A that results in Ax = 0. Signup and view all the answers

What is Row Space C(A^T)?

The span of the rows in A. Signup and view all the answers

What is the Left Null Space N(A^T)?

Set of all solutions that x^T A = 0. Signup and view all the answers

What does transpose mean?

Flipping the matrix across the main diagonal. Signup and view all the answers

How do you test for inverse of a matrix A?

Must have a pivot for every row/column. Signup and view all the answers

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.