An m×n upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer
An m×n upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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An
m×n
upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer. |
Transcribed Image Text:An mxn upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When
is a square upper triangular matrix invertible? Justify your answer.
3 4 7 4
0 1 4 6
0 0 2 8
0 0 0 1
Choose the correct answer below.
O A. A square upper triangular matrix is invertible when all entries on the main diagonal are ones. If any entry on the main diagonal is not one, then
the equation Ax = b, where A is an nxn square upper triangular matrix, has no solution for at least one b in R".
OB.
A square upper triangular matrix is invertible when the matrix is equal to its own transpose. For such a matrix A, A = A' means that the
equation Ax = b has at least one solution for each b in R".
O C. A square upper triangular matrix is invertible when all entries above the main diagonal are zeros as well. This means that the matrix is row
equivalent to the nxn identity matrix.
O D. A square upper triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are
nonzero, then the nxn matrix has n pivot positions.
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