$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 $ det (W) =

Question

BlackTom AI · Accepted Answer

We are given the 2x2 matrix W = [[0.6, 0.5], [0.4, 0.5]] and asked for det(W).
First, recall the formula for the determinant of a 2x2 matrix [[a, b], [c, d]]: det = ad − bc.

Option a: -0.5. If we plug a = 0.6, b = 0.5, c = 0.4, d = 0.5 into ad − bc, we get (0.6)(0.5) − (0.5)(0.4) = 0.30 − 0.20 = 0.10, not -0.5. This shows this choice is incorrect.

Option b: 0.5. Using the same calculation ad − bc = 0.30 − 0.20 = 0.10, we see the actual determinant is 0.10, not 0.5. Therefore this option is also incorrect.

Option c: 0.1. This matches the computed value: 0.30 − 0.20 = 0.10, which is 0.1 when written in decimal form. Hence this option is consistent with the determinant formula.

Option d: 0.2. Since the exact calculation yields 0.10, not 0.2, this option is inaccurate.

Option e: not defined. The determinant of a 2x2 matrix is always defined (provided the entries are real numbers, which they are here), so claiming 'not defined' is incorrect.

Summarizing the checks, only option c yields the correct determinant value 0.1, while all other options either miscalculated the result or misunderstand the determinant's existence for a 2x2 matrix.

MCD1550 / MCD2140 - T1 - 2025 TEST 2 (Week 6)

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