Questions and Answers
What are the two conditions for a square matrix to be singular?
Its RREF contains a zero row and its det = 0.
What is an elementary matrix?
A matrix which differs from the identity matrix by one single elementary row operation.
All elementary matrices are non-singular.
True
If A is a matrix, then what is the determinant of A transpose?
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How does interchanging rows affect det(A)?
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How does multiplying a scalar k to a row affect det(A)?
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What happens to det(A) if we multiply a scalar k to a row and add it to another row?
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What does it mean for matrices A and B to be row equivalent?
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If E is an elementary matrix, then det(EA) = ...
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If matrix A is non-singular, what can be said about det(A)?
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What is the relationship between det(AB) and the determinants of A and B?
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If det(A) does not equal zero, is matrix A linearly independent or dependent?
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What is a symmetric matrix?
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What is a skew symmetric matrix?
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What is Cramer’s rule?
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What are the properties of matrix addition?
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What are the properties of matrix multiplication?
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What are the properties of the transpose of a matrix?
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What are three differences between matrix multiplication and multiplication of real numbers?
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If A and B are nonsingular nxn matrices, then what can be said about AB?
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If A is nonsingular, what can be said about the transpose of A and the inverse of A^T?
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If A is a matrix, then how do you express the determinant of A?
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What is the relationship between det(A^-1) and det(A)?
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What defines a subspace W of a vector space V?
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What can be said about the span of a set of vectors S in V?
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When are vectors V1, ..., Vr in a vector space said to be linearly dependent?
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What condition indicates that a set S of n vectors in Rn is linearly independent?
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If dimV = n, what can be said about the vectors in V?
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What can be said about a linearly independent set of vectors S in a finite-dimensional vector space V?
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If S is a linearly independent set of vectors in an n-dimensional vector space V, what can be concluded?
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If S spans V, what can be inferred?
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What does it mean if spanS = V for a finite subset S of vector space V?
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How can we find the basis for V if V = { x | Ax = 0 , A exists in Mmn } is a subspace for Rn?
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Study Notes
Matrix Definitions and Properties
- A square matrix is singular if its reduced row echelon form (rref) contains a zero row or its determinant equals zero.
- An elementary matrix results from applying a single elementary row operation to the identity matrix.
Determinant Properties
- All elementary matrices are non-singular, meaning they have a non-zero determinant.
- The determinant of a matrix remains unchanged when transposed.
- Interchanging two rows of a matrix negates its determinant (det(A) becomes -det(A)).
- Multiplying a row by a scalar ( k ) scales the determinant by ( k ) (det(A) becomes k * det(A)).
- Adding a multiple of one row to another row leaves the determinant unchanged.
- If ( A ) and ( B ) are matrices, then det(AB) = det(A) * det(B).
- If det(A) is non-zero, matrix ( A ) is linearly independent.
Matrix Types
- A symmetric matrix satisfies the property ( A^T = A ).
- A skew-symmetric matrix satisfies ( A^T = -A ).
Cramer's Rule
- Used to solve systems of linear equations with a non-zero determinant.
Properties of Matrix Operations
- For matrix addition:
- A + B = B + A (commutative),
- A + (B + C) = (A + B) + C (associative),
- There exists a zero matrix such that A + 0 = A,
- Each matrix has an additive inverse where A + (-A) = 0.
- For matrix multiplication:
- A(BC) = (AB)C (associative),
- (A + B)C = AC + BC (distributive),
- C(A + B) = CA + CB (distributive).
Properties of Transpose
- A double transpose returns the original matrix: ( (A^T)^T = A ).
- The transpose of a sum of matrices equals the sum of their transposes.
- The transpose of a product reverses the order: ( (AB)^T = B^T A^T ).
- The transpose of a scalar multiplied by a matrix is the scalar multiplied by the transpose of the matrix: ( (rA)^T = rA^T ).
Linear Independence and Basis
- Vectors ( V_1, \ldots, V_r ) are linearly dependent if there are constants ( a_1, a_2, \ldots, a_r ) (not all zero) such that ( a_1V_1 + a_2V_2 + \cdots + a_rV_r = 0 ).
- A set of ( n ) vectors in ( \mathbb{R}^n ) is linearly independent if the determinant of their corresponding matrix ( A ) is non-zero.
- In an ( n )-dimensional vector space ( V ):
- Any subset of more than ( n ) vectors must be linearly dependent.
- Any subset with fewer than ( n ) vectors cannot span ( V ).
- A finite set ( S ) of vectors in ( V ) with span equal to ( V ) has a maximal independent subset ( T ) that forms a basis for ( V ).
- If ( S ) spans ( V ) or is a linearly independent set, it is a basis for ( V ).
Finding the Basis
- To find the basis for ( V ) defined by ( V = {x | Ax = 0} ):
- Create the augmented matrix [A|0],
- Transform it into reduced row echelon form (rref) to identify non-zero rows, which provide the basis.
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Test your knowledge of key concepts in linear algebra with these flashcards from Purdue's MA 265 course. Topics include singular matrices, elementary matrices, and determinants. Perfect for quick review and exam preparation.