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What is the cofactor of a matrix element ai,j?
What is the cofactor of a matrix element ai,j?
What is the determinant of an upper or lower triangular matrix?
What is the determinant of an upper or lower triangular matrix?
When a matrix has a row or column of zeros, what is the value of its determinant?
When a matrix has a row or column of zeros, what is the value of its determinant?
What happens to the determinant of a matrix when two adjacent rows are interchanged?
What happens to the determinant of a matrix when two adjacent rows are interchanged?
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Which expansion can be used to compute the determinant of a matrix?
Which expansion can be used to compute the determinant of a matrix?
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If the matrix A is a 4x4 lower triangular matrix, how is det(A) calculated?
If the matrix A is a 4x4 lower triangular matrix, how is det(A) calculated?
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In cofactor expansion along row i, which term contributes to the determinant if ai,j is zero?
In cofactor expansion along row i, which term contributes to the determinant if ai,j is zero?
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How does the determinant behave if a row of a matrix is multiplied by a scalar?
How does the determinant behave if a row of a matrix is multiplied by a scalar?
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What happens to the determinant of a matrix when two rows of it are identical?
What happens to the determinant of a matrix when two rows of it are identical?
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What is the relationship between the determinant of a matrix and its invertibility?
What is the relationship between the determinant of a matrix and its invertibility?
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If a row of a matrix is multiplied by a scalar $ ext{λ}$, how does this affect the determinant?
If a row of a matrix is multiplied by a scalar $ ext{λ}$, how does this affect the determinant?
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In the reduction of a matrix to echelon form, if a matrix does not have all pivots that are non-zero, what can be concluded?
In the reduction of a matrix to echelon form, if a matrix does not have all pivots that are non-zero, what can be concluded?
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What is the determinant of $ ext{λA}$, where A is a square matrix and λ is a scalar?
What is the determinant of $ ext{λA}$, where A is a square matrix and λ is a scalar?
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How can the determinant of a matrix be efficiently computed for large matrices?
How can the determinant of a matrix be efficiently computed for large matrices?
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Which of the following statements is true regarding the effect of adding a multiple of one row to another row on the determinant?
Which of the following statements is true regarding the effect of adding a multiple of one row to another row on the determinant?
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Which of the following best describes the theorem that relates the determinant and invertibility of a square matrix?
Which of the following best describes the theorem that relates the determinant and invertibility of a square matrix?
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What happens to the determinant when two adjacent rows of a matrix are interchanged?
What happens to the determinant when two adjacent rows of a matrix are interchanged?
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If two rows of a matrix are identical, what is the value of the determinant?
If two rows of a matrix are identical, what is the value of the determinant?
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How does the determinant change when a row of a matrix is expressed as the sum of two other rows?
How does the determinant change when a row of a matrix is expressed as the sum of two other rows?
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What is the effect on the determinant when a multiple of one row is added to another row in a matrix?
What is the effect on the determinant when a multiple of one row is added to another row in a matrix?
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Which of the following statements is true regarding the determinant of a matrix?
Which of the following statements is true regarding the determinant of a matrix?
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What is the relationship between the determinants of square matrices A and B when B is obtained by interchanging two non-adjacent rows of A?
What is the relationship between the determinants of square matrices A and B when B is obtained by interchanging two non-adjacent rows of A?
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In a (3x3) matrix, when a row is formed as a linear combination of two other rows, what conclusion can be reached about the determinant?
In a (3x3) matrix, when a row is formed as a linear combination of two other rows, what conclusion can be reached about the determinant?
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What can be concluded if a matrix A has all its rows twice and interchanged?
What can be concluded if a matrix A has all its rows twice and interchanged?
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Study Notes
Properties of Determinants
- Expansion with respect to any row/column: A determinant can be computed using a cofactor expansion with respect to any row or column. The result is always the same.
- Cofactor: The number Cij = (-1)i+j det(Aij) is called the (i,j) cofactor of matrix A. Aij is the submatrix obtained from A by removing row i and column j.
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Zero row/column: If a matrix has a zero row or column, its determinant is zero.
- Proof: Using the cofactor expansion formula, if a row has all zeros, the determinant of the matrix is zero.
Upper/Lower Triangular Matrices
- Determinant of triangular matrices: If A is an upper or lower triangular matrix, the determinant of A equals the product of its diagonal entries.
- Example (4x4): For a lower triangular 4x4 matrix, det(A) = a11 * a22 * a33 * a44.
- General case: A similar proof holds for n x n triangular matrices.
Row/Column Interchange
- Effect on determinant: Interchanging any two rows or columns of a matrix changes the sign of its determinant.
- Example (2x2): If A = [a11 a12; a21 a22] then det(A) = a11 a22 - a12 a21. Interchanging rows of A yields B = [a21 a22; a11 a12] and det(B) = a21 a12 - a22 a11 = - det(A).
Two identical rows/columns
- Determinant is zero: If a matrix has two identical rows or columns, its determinant is zero.
Linearity of Determinants (with respect to rows)
- Sum of row vectors: If the first row of a matrix is a sum of two row vectors, the determinant is the sum of the determinants of the matrices where the first row is each individual row vector.
Adding a row to another row in a Matrix
- No effect on determinant: Adding a multiple of one row to another row in a matrix does not change the determinant.
- Example: If you add a multiple of row i to row j, the determinant is unchanged.
Scalar multiplication of a row/column
- Effect on determinant: If a row or column of a matrix is multiplied by a scalar, the determinant is also multiplied by that scalar.
- Example: If a row is multiplied by alpha, the determinant is multiplied by alpha.
Invertibility and Determinants
- A matrix is invertible if and only if its determinant is non-zero: This is a crucial property linking invertibility and determinants.
- Singular matrix: A matrix with a determinant of 0 is a singular matrix (not invertible).
Determinant Calculation Methods
- Cofactor expansion: A method for calculating determinants, but it can be computationally expensive for large matrices.
- Gaussian elimination: A process that reduces a matrix to a simpler form for computing a determinant more efficiently, especially for large matrices. This is generally the preferred method in practice.
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Description
This quiz explores key concepts related to the properties of determinants and their calculation using cofactor expansion. It also covers the determinants of upper and lower triangular matrices, providing essential examples and proofs. Test your understanding of these important mathematical principles!
