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Hwk 7: Matrix Inverses
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Question 6, 2.3.22 >
An $m \times n$ lower triangular matrix is one whose entries above the main diagonal are zeros, as is shown in the matrix to the right. When is a square lower triangular matrix invertible? Justify your answer.
Choose the correct answer below.
A. A square lower triangular matrix is invertible when all entries below the main diagonal are zeros as well. This means that the matrix is row equivalent to the $n \times n$ identity matrix.
B. A square lower 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 $n \times n$ matrix has $n$ pivot positions.
C. A square lower 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 $n \times n$ square lower triangular matrix, has no solution for at least one $b$ in $\mathbb{R}^n$.
D. A square lower triangular matrix is invertible when the matrix is equal to its own transpose. For such a matrix $A$, $A = A^T$ means that the equation $Ax = b$ has at least one solution for each $b$ in $\mathbb{R}^n$
