Questions and Answers
The determinant of a matrix is a matrix of the same size.
False
Det ( \begin{pmatrix} a & b \ c & d \end{pmatrix} ) = ad + bc.
False
If the determinant of a 2 x 2 matrix equals zero, then the matrix is invertible.
False
If a 2 x 2 matrix is invertible, then its determinant equals zero.
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If B is a matrix obtained by multiplying each entry of some row of a 2 × 2 matrix A by the scalar k, then det B = k det A.
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For n ≥ 2, the (i, j)-cofactor of an n × n matrix A is the determinant of the (n − 1) × (n − 1) matrix obtained by deleting row i and column j from A.
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For n ≥ 2, the (i, j)-cofactor of an n × n matrix A equals ((-1)^{i+j}) times the determinant of the (n − 1) × (n − 1) matrix obtained by deleting row i and column j from A.
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The determinant of an n × n matrix can be evaluated by a cofactor expansion along any row.
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Cofactor expansion is an efficient method for evaluating the determinant of a matrix.
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The determinant of a matrix with integer entries must be an integer.
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The determinant of a matrix with positive entries must be positive.
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If some row of a square matrix consists only of zero entries, then the determinant of the matrix equals zero.
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An upper triangular matrix must be square.
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A matrix in which all the entries to the left and below the diagonal entries equal zero is called a lower triangular matrix.
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A 4 × 4 upper triangular matrix has at most 10 nonzero entries.
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The transpose of a lower triangular matrix is an upper triangular matrix.
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The determinant of an upper triangular n × n matrix or a lower triangular n × n matrix equals the sum of its diagonal entries.
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The determinant of ( I_n ) equals 1.
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The area of the parallelogram determined by u and v is det [u v].
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Study Notes
Determinants and Matrix Properties
- A determinant is not a matrix; it is a scalar value that can be calculated from a square matrix.
- The determinant formula for a 2x2 matrix does not equal ad + bc; the correct formula is ad - bc.
- A 2x2 matrix is not invertible if its determinant equals zero; a determinant of zero indicates the matrix is singular and cannot be inverted.
- For an invertible 2x2 matrix, the determinant must be non-zero, not equal to zero.
Cofactor Expansion
- If a matrix B is derived from matrix A by multiplying every entry of a row by a scalar k, then the relationship holds that det B = k * det A.
- The (i, j)-cofactor of an n x n matrix is defined as (−1)^(i+j) multiplied by the determinant of the resulting (n − 1) x (n − 1) matrix after removing the ith row and jth column.
- The determinant can be computed using cofactor expansion along any row of the matrix.
Characteristics of Determinants
- Cofactor expansion is not necessarily an efficient method for computing a matrix's determinant; efficiency may vary based on the matrix structure.
- A matrix with integer entries guarantees that its determinant is also an integer.
- A matrix with positive entries does not ensure that its determinant will be positive; determinants can be negative regardless of the sign of the entries.
Special Matrix Conditions
- If any row in a square matrix is entirely composed of zero entries, then the determinant of that matrix is zero.
- An upper triangular matrix must be square, allowing for defined rows and columns corresponding to the same size.
Types of Triangular Matrices
- A matrix with zeros to the left and below the diagonal is referred to as an upper triangular matrix, while a lower triangular matrix has zeros above the diagonal.
- A 4x4 upper triangular matrix can have at most 10 non-zero entries, as it has four diagonal entries.
- The transpose of a lower triangular matrix results in an upper triangular matrix.
Additional Determinant Properties
- The determinant of the identity matrix (In) is always 1.
- The statement regarding the diagonal sum of upper or lower triangular matrices being equal to their determinant is false; the determinant is actually the product of the diagonal entries, not the sum.
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Description
Test your understanding of determinants and their properties in matrices. The quiz covers key concepts including determinant calculation, cofactor expansion, and characteristics of invertible matrices. Challenge yourself to apply these ideas to various matrix problems!
