Podcast
Questions and Answers
The determinant of a matrix is a matrix of the same size.
The determinant of a matrix is a matrix of the same size.
False (B)
Det ( \begin{pmatrix} a & b \ c & d \end{pmatrix} ) = ad + bc.
Det ( \begin{pmatrix} a & b \ c & d \end{pmatrix} ) = ad + bc.
False (B)
If the determinant of a 2 x 2 matrix equals zero, then the matrix is invertible.
If the determinant of a 2 x 2 matrix equals zero, then the matrix is invertible.
False (B)
If a 2 x 2 matrix is invertible, then its determinant equals zero.
If a 2 x 2 matrix is invertible, then its determinant equals zero.
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.
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.
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.
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.
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.
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.
The determinant of an n × n matrix can be evaluated by a cofactor expansion along any row.
The determinant of an n × n matrix can be evaluated by a cofactor expansion along any row.
Cofactor expansion is an efficient method for evaluating the determinant of a matrix.
Cofactor expansion is an efficient method for evaluating the determinant of a matrix.
The determinant of a matrix with integer entries must be an integer.
The determinant of a matrix with integer entries must be an integer.
The determinant of a matrix with positive entries must be positive.
The determinant of a matrix with positive entries must be positive.
If some row of a square matrix consists only of zero entries, then the determinant of the matrix equals zero.
If some row of a square matrix consists only of zero entries, then the determinant of the matrix equals zero.
An upper triangular matrix must be square.
An upper triangular matrix must be square.
A matrix in which all the entries to the left and below the diagonal entries equal zero is called a lower triangular matrix.
A matrix in which all the entries to the left and below the diagonal entries equal zero is called a lower triangular matrix.
A 4 × 4 upper triangular matrix has at most 10 nonzero entries.
A 4 × 4 upper triangular matrix has at most 10 nonzero entries.
The transpose of a lower triangular matrix is an upper triangular matrix.
The transpose of a lower triangular matrix is an upper triangular matrix.
The determinant of an upper triangular n × n matrix or a lower triangular n × n matrix equals the sum of its diagonal entries.
The determinant of an upper triangular n × n matrix or a lower triangular n × n matrix equals the sum of its diagonal entries.
The determinant of ( I_n ) equals 1.
The determinant of ( I_n ) equals 1.
The area of the parallelogram determined by u and v is det [u v].
The area of the parallelogram determined by u and v is det [u v].
Flashcards are hidden until you start studying
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
