31. For which values of the constants a, b, and c is the fol- lowing matrix...
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31. For which values of the constants a, b, and c is the fol- lowing matrix invertible? 32. Find all matrices 0 -b a b 0 C 0 such that ad-be = 1 and 33. Consider the matrices of the form A = where a and b are arbitrary constants. For which val- ues of a and b is A = A? 34. Consider the diagonal matrix a 0 0 A=0 b 0 00 C A = . a a. For which values of a, b, and c is A invertible? If it is invertible, what is A-12 b. For which values of the diagonal elements is a dia- gonal matrix (of arbitrary size) invertible? 35. a. Consider the upper triangular 3 x 3 matrix a b C 0 d e 14 0 0f For which values of a, b, c, d, e, and fis invert- ible? b. More generally, when is an upper triangular matrix (of arbitrary size) invertible? c. If an upper triangular matrix is invertible, is its in- verse an upper triangular matrix as well? d. When is a lower triangular matrix invertible? 36. To determine whether a square matrix A is invertible, it is not always necessary to bring it into reduced row- echelon form. Instead, reduce A to (upper or lower) tri- angular form, using elementary row operations. Show that A is invertible if (and only if) all entries on the di- agonal of this triangular form are nonzero. 31. For which values of the constants a, b, and c is the fol- lowing matrix invertible? 32. Find all matrices 0 -b a b 0 C 0 such that ad-be = 1 and 33. Consider the matrices of the form A = where a and b are arbitrary constants. For which val- ues of a and b is A = A? 34. Consider the diagonal matrix a 0 0 A=0 b 0 00 C A = . a a. For which values of a, b, and c is A invertible? If it is invertible, what is A-12 b. For which values of the diagonal elements is a dia- gonal matrix (of arbitrary size) invertible? 35. a. Consider the upper triangular 3 x 3 matrix a b C 0 d e 14 0 0f For which values of a, b, c, d, e, and fis invert- ible? b. More generally, when is an upper triangular matrix (of arbitrary size) invertible? c. If an upper triangular matrix is invertible, is its in- verse an upper triangular matrix as well? d. When is a lower triangular matrix invertible? 36. To determine whether a square matrix A is invertible, it is not always necessary to bring it into reduced row- echelon form. Instead, reduce A to (upper or lower) tri- angular form, using elementary row operations. Show that A is invertible if (and only if) all entries on the di- agonal of this triangular form are nonzero.
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Related Book For
Differential Equations and Linear Algebra
ISBN: 978-0131860612
2nd edition
Authors: Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West
Posted Date:
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