$U = \left[ {\begin{array}{*{20}{c}}1&1&0\2&3&4\end{array}} \right]\,\,\;\quad V = \left[ {\begin{array}{*{20}{c}}3\1\end{array}} \right]\,\,\;\quad W = \left[ {\begin{array}{*{20}{c}}{0.6}&{0.5}\{0.4}&{0.5}\end{array}} \right]\;\,\,\;X = \left[ {\begin{array}{*{20}{c}}3&5\4&6\end{array}} \right]\quad \,\, \ Y = \left[ {\begin{array}{*{20}{c}}1&3&2\0&5&6\0&0&1\end{array}} \right]\;\quad $ The upper triangular matrix is:

Question

BlackTom AI · Accepted Answer

The task asks which given matrix is upper triangular. Let's examine each option carefully and note both shape and entries.
Option by option analysis:
- U = [ [1, 1, 0], [2, 3, 4] ]. This matrix is 2x3, i.e., not square. Standard definition of an upper triangular matrix requires a square matrix, where all entries below the main diagonal are zero. Here there are nonzero entries below the diagonal (the 2 in row 2, column 1), so U is not upper triangular.
- V = [ [3], [1] ]. This is a 2x1 column vector. While some definitions discuss triangularity for square matrices, a non-square shape cannot be upper triangular under the conventional definition. Since V is not square, it cannot be upper triangular.
- W = [ [0.6, 0.5], [0.4, 0.5] ]. This 2x2 matrix has a nonzero entry below the main diagonal (0.4 in row 2, column 1) and a nonzero above the diagonal (0.5 in row 1, column 2). Because entries both below and above the diagonal are nonzero, W is neither upper nor lower triangular.
- X = [ [3, 5], [4, 6] ]. This 2x2 matrix has a nonzero entry below the main diagonal (4 in row 2, column 1). That disqualifies it from being upper triangular, since an upper triangular matrix must have zeros below the diagonal. It is not upper triangular.
- Y = [ [1, 3, 2], [0, 5, 6], [0, 0, 1] ]. This 3x3 matrix has zeros below the main diagonal: positions (2,1) and (3,1) and (3,2) are all zero. This matches the standard definition of an upper triangular matrix. Moreover, it is square, which fits the usual requirement for triangularity.
In summary, among all given matrices, only Y satisfies the criteria for being upper triangular, as it is a square matrix with zeros below the diagonal. The other matrices fail due to non-square shapes or having nonzero entries below (and/or above) the diagonal. Consequently, the correct choice is Y.

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