Questions and Answers
What is the effect of applying $r_i \rightarrow a r_i$ on the determinant?
What is the first step in finding the inverse of a 3x3 matrix?
The pattern for flipping signs when computing the minors of a matrix A is: ______
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- +, - + -, + - +
Which of the following is true about linear transformations?
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What does nullity equate to?
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What is the rank of a matrix?
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If $\lambda$ is an eigenvalue for matrix A, then $-\lambda$ is also an eigenvalue for A.
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How do you test if a vector $v$ is in the span of vectors $w_1, \ldots, w_k$?
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What is the determinant of the matrix representing the area of a triangle with corners $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$?
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Study Notes
Effect of Elementary Row Operations on Determinant
- Multiplying a row by a scalar ( a ) multiplies the determinant by ( a ).
- Adding a multiple of one row to another row does not affect the determinant.
- Swapping two rows changes the sign of the determinant.
Inverse of 3x3 Matrix
- Calculate the determinant of matrix ( A ).
- Compute minors of ( A ) for each element.
- Apply sign pattern:
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- Transpose the matrix of cofactors to obtain the adjugate.
- Divide the adjugate by the determinant to get the inverse.
Find the Adjugate (3x3)
- Construct the matrix of minors for matrix ( A ).
- Apply the alternating sign pattern to generate the matrix of cofactors.
- Transpose the matrix of cofactors to find the adjugate.
Linear Transformation
- Must maintain the form ( Ax ) and cannot include exponents or constants.
Nullity Fallacies
- Nullity equals the number of nonzero rows in row echelon form.
- Nullity does not equal the dimension of the column space.
- Nullity is not the same as the nullity of the transposed matrix.
Square Matrix Fallacies
- If matrix ( B ) is obtained from ( A ) through row operations, both share the same eigenvalue ( \lambda ).
- Determinant of the sum ( \text{det}(A+B) ) does not equal ( \text{det}(A) + \text{det}(B) ).
- If ( A ) and ( B ) are both nonsingular, their sum ( A+B ) may not be nonsingular.
Real Square Matrix Truths
- If ( \lambda ) is an eigenvalue of ( A ), then ( -\lambda ) is an eigenvalue of ( -A ).
- Any eigenvector ( v ) for ( A ) is also an eigenvector for ( 2A ).
Linear Independence
- A set of vectors is linearly independent if each column in reduced row echelon form (Rref) contains a leading one as its only nonzero entry.
The Following Are the Same
- The nullspace of matrix ( A ) is identical to the solution space of ( AX=0 ) and the orthogonal complement of the row space of ( A ).
Rank
- Defined as the number of leading 1's present in the Rref of matrix ( A ).
Nullity
- The number of columns in ( A ) that do not have leading 1’s in Rref represents the nullity.
To Test Whether a Vector ( v ) is in the Span of Vectors ( w_1, \ldots, w_k )
- Formulate the matrix ( A ) with ( w_1, \ldots, w_k ) as columns and check if ( AX=v ) maintains consistency.
To Test Whether Vectors ( w_1, \ldots, w_k ) Span the Vector Space ( V )
- Create matrix ( A ) with ( w_1, \ldots, w_k ) as columns and perform Rref. If every row contains a leading 1, then they span ( V ).
Square ( n \times n ) Matrix ( A )
- An invertible square matrix has a non-zero determinant, rank ( n ), linearly independent rows and columns, yielding exactly one solution for all ( b ), and a nullity of zero.
Determinant of ( kA )
- The determinant of matrix ( A ) multiplied by ( k ) is ( \text{det}(A) \times k^n ), where ( n ) is the number of rows in ( A ).
Area of a Triangle with Corners ((x_1,y_1), (x_2,y_2), (x_3,y_3))
- The area can be calculated using the determinant of the matrix: [ \begin{bmatrix} 1 & 1 & 1 \ x_1 & x_2 & x_3 \ y_1 & y_2 & y_3 \end{bmatrix} ]
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Test your knowledge of linear algebra concepts with these flashcards focused on elementary row operations and the inverse of 3x3 matrices. These cards provide key definitions and insights crucial for mastering linear algebra at Purdue University.
