Given the matrices $$ A= \left[\begin{array}{ccc} -1&0&3 \ -2&4&6 \end{array}\right] $$ and $$ B= \left[\begin{array}{cccc} -1&3&5&0 \ 1&4&2&1 \end{array}\right] $$ if $ AX = B $ the order of matrix $X$ is

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BlackTom AI · Accepted Answer

To determine the size of X we use the rule for matrix multiplication: if A is of size m×n and X is of size p×q, then AX is defined only when n = p, and the result has size m×q.

Option a: 3x4. Here X would be 3×4. Since A is 2×3, the inner dimension matches (3), so the product AX is defined and would have size 2×4, which matches B's size. This is consistent with AX = B.

Option b: 4x3. If X were 4×3, then A (2×3) cannot multiply X (4×3) because the inner dimensions would be 3 and 4, which do not match. Therefore AX is undefined in this case.

Option c: 3x2. If X were 3×2, the product AX would have size 2×2 (since A is 2×3 and the result has the outer dimensions from A and X: m×q = 2×2). But B is 2×4, not 2×2, so this option cannot satisfy AX = B.

Option d: 2x3. If X were 2×3, the product AX would have size 2×3. Again, this does not match B’s size of 2×4, so AX ≠ B in this case.

In summary, only a. 3x4 yields a defined product AX with the correct result shape to equal B (2×4). The other options fail either the dimension compatibility requirement or the resulting dimension requirement for AX to equal B.

Given the matrices $$ A= \left[\begin{array}{ccc} -1&0&3 \\ -2&4&6 \end{array}\right] $$ and $$ B= \left[\begin{array}{cccc} -1&3&5&0 \\ 1&4&2&1 \end{array}\right] $$ if \( AX = B \) the order of matrix \(X\) is单项选择题

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