If $$ \left [\begin{matrix}0 & 0\0 & -2\end{matrix}\right ] X + \left [\begin{matrix}-2 & 0\2 & -3\end{matrix}\right ] = \left [\begin{matrix}-3 & 1\-3 & 0\end{matrix}\right ] X, $$ which of the following matrices is $X$?

Question

BlackTom AI · Accepted Answer

We start by identifying the matrices involved in the equation. Let A = [[0, 0], [0, -2]], B = [[-2, 0], [2, -3]], and C = [[-3, 1], [-3, 0]]. The equation is A X + B = C X.

Rearranging terms gives (A - C) X = -B. Compute A - C entrywise: A - C = [[0 - (-3), 0 - 1], [0 - (-3), -2 - 0]] = [[3, -1], [3, -2]]. Therefore we have the matrix equation [[3, -1], [3, -2]] X = -B, and -B = [[2, 0], [-2, 3]].

Let X be a 2x2 matrix with entries X = [[x11, x12], [x21, x22]]. Multiplying M X with M = [[3, -1], [3, -2]] gives:
- First row: [3, -1] times X yields [3x11 - x21, 3x12 - x22]. This must equal the first row of -B, which is [2, 0]. So we have the equations 3x11 - x21 = 2 and 3x12 - x22 = 0.
- Second row: [3, -2] times X yields [3x11 - 2x21, 3x12 - 2x22]. This must equal the second row of -B, which is [-2, 3]. So we have 3x11 - 2x21 = -2 and 3x12 - 2x22 = 3.

Now solve the system for the first column (x11, x21):
From 3x11 - x21 = 2, we get x21 = 3x11 - 2. Substitute into 3x11 - 2x21 = -2: 3x11 - 2(3x11 - 2) = -2 → 3x11 - 6x11 + 4 = -2 → -3x11 = -6 → x11 = 2. Then x21 = 3(2) - 2 = 4.
Proceed to the second column (x12, x22):
From 3x12 - x22 = 0, we get x22 = 3x12. Substitute into 3x12 - 2x22 = 3: 3x12 - 2(3x12) = 3 → 3x12 - 6x12 = 3 → -3x12 = 3 → x12 = -1. Then x22 = 3(-1) = -3.

Assembling the solution, X = [[2, -1], [4, -3]]. This matches the provided option: [[2, -1], [4, -3]].

If \[ \left [\begin{matrix}0 & 0\\0 & -2\end{matrix}\right ] X + \left [\begin{matrix}-2 & 0\\2 & -3\end{matrix}\right ] = \left [\begin{matrix}-3 & 1\\-3 & 0\end{matrix}\right ] X, \] which of the following matrices is \(X\)?单项选择题

类似问题

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