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36. In a symmetric n x n matrix, the (i,j)- and (j, i)-entries are equal for all i = 1,2,.n and j = 1,2,.,n.

36. In a symmetric n x n matrix, the (i,j)- and (j, i)-entries are equal for all i = 1,2,.n and j = 1,2,.,n.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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36. In a symmetric n x n matrix, the (i,j)- and (j, i)-entries are equal for all i = 1,2,...n and j = 1,2,..,n. 37. The determinant of a matrix is a matrix of the same size. а 38. det = ad + bc. 39. If the determinant of a 2 x 2 matrix equals zero, then the matrix is invertible. 40. If a 2 x 2 matrix is invertible, then its determinant equals zero. 41. 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. 42. For n > 2, the (i,j)-cofactor of an n x n matrix A is the determinant of the (n – 1) × (n – 1) matrix obtained by deleting row i and column j from A. 43. For n > 2, the (i,j)-cofactor of an n x 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. 44. The determinant of an n x n matrix can be evaluated by a cofactor expansion along any row. 45. Cofactor expansion is an efficient method for evaluating the determinant of a matrix. 46. The determinant of a matrix with integer entries must be an integer.
Transcribed Image Text:36. In a symmetric n x n matrix, the (i,j)- and (j, i)-entries are equal for all i = 1,2,...n and j = 1,2,..,n. 37. The determinant of a matrix is a matrix of the same size. а 38. det = ad + bc. 39. If the determinant of a 2 x 2 matrix equals zero, then the matrix is invertible. 40. If a 2 x 2 matrix is invertible, then its determinant equals zero. 41. 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. 42. For n > 2, the (i,j)-cofactor of an n x n matrix A is the determinant of the (n – 1) × (n – 1) matrix obtained by deleting row i and column j from A. 43. For n > 2, the (i,j)-cofactor of an n x 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. 44. The determinant of an n x n matrix can be evaluated by a cofactor expansion along any row. 45. Cofactor expansion is an efficient method for evaluating the determinant of a matrix. 46. The determinant of a matrix with integer entries must be an integer.
47. The determinant of a matrix with positive entries must be positive. 48. If some row of a square matrix consists only of zero entries, then the determinant of the matrix equals zero. 49. An upper triangular matrix must be square. 50. A matrix in which all the entries to the left and below the diagonal entries equal zero is called a lower triangular matrix. 51. A 4 x 4 upper triangular matrix has at most 10 nonzero entries. 52. The transpose of a lower triangular matrix is an upper triangular matrix. 53. The determinant of an upper triangular n x n matrix or a lower triangular n xn matrix equals the sum of its diag- onal entries. 54. The determinant of I, equals 1.
Transcribed Image Text:47. The determinant of a matrix with positive entries must be positive. 48. If some row of a square matrix consists only of zero entries, then the determinant of the matrix equals zero. 49. An upper triangular matrix must be square. 50. A matrix in which all the entries to the left and below the diagonal entries equal zero is called a lower triangular matrix. 51. A 4 x 4 upper triangular matrix has at most 10 nonzero entries. 52. The transpose of a lower triangular matrix is an upper triangular matrix. 53. The determinant of an upper triangular n x n matrix or a lower triangular n xn matrix equals the sum of its diag- onal entries. 54. The determinant of I, equals 1.
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