Let A be an n×n matrix. Mark each statement as true or false. Justify each answer. a. An n×n determinant is define...
Let A be an n×n matrix. Mark each statement as true or false. Justify each answer.
|
a. |
An n×n determinant is defined by determinants of (n−1)×(n−1) submatrices. |
|
b. |
The (i,j)-cofactor of a matrix A is the matrix obtained by deleting from A its I’th row and j’th column. |
a. Choose the correct answer below.
A. The statement is false. Although determinants of (n−1)×(n−1)submatrices can be used to find n×n determinants,they are not involved in the definition of n×n determinants.
B. The statement is true. The determinant of an n×n
matrix A can be computed by a cofactor expansion across any row or down any column. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size
(n−1)×(n−1).
C. The statement is true. The determinant of an n×n
matrix A can be computed by a cofactor expansion along either diagonal. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size
(n−1)×(n−1).
D. The statement is false. An n×n determinant is defined by determinants of
(n−1)×(n−1)submatrices only when n>3. Determinants of 1×1, 2×2,and 3×3 matrices are defined separately.
b. Choose the correct answer below.
A. The statement is false. The (i,j)-cofactor of a matrix A is the matrix Aij
obtained by deleting from A its jth row and ith column.
B. The statement is false. The (i,j)-cofactor of A is the number
Cij=det(Aij),where Aij
is the submatrix obtained by deleting from A its ith row and jth column.
C. The statement is true. It is the definition of the (i,j)-cofactor of a matrix A.
D. The statement is false. The (i,j)-cofactor of A is the number Cij=(−1)i+jdetAij, where Aij
is the submatrix obtained by deleting from A its ith row and jth column.
Homework Answers
a. FALSE: The statement is false. Although determinants of (n−1)×(n−1)submatrices can be used to find n × n determinants, they are not involved in the definition of n × n determinants. Option A is the correct answer
b. FALSE: The statement is false. The (i,j)-cofactor of A is the number Cij=(−1)i+j det Aij, where Aij
is the submatrix obtained by deleting from A its ith row and jth column. Option D is the correct answer.
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