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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Question restatement: We are given A as a 2×3 matrix and B as a 2×4 matrix, with the relation AX = B. We need the dimensions of X.
Step 1: Identify the dimensions of A and B. Matrix A is 2 rows by 3 columns (2×3). Matrix B is 2 rows by 4 columns (2×4).
Step 2: Use the rule for matrix multiplication dimension compatibility. 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 AX will have size m×q.
Step 3: Apply to our case. Here, A is 2×3, so n = 3. Therefore X must have p = 3 (its number of rows) to be conformable.
Step 4: Determine the resulting number of columns of AX. Since AX = B has 2 rows and 4 columns, the result must be 2×4. This means q = 4, so X must have 4 columns.
Conclusion from the dimensionality: X must be a 3×4 matrix to satisfy AX = B.
Additional note: The question presents an option labeled as d. 3x4, which matches the required dimensions. While this analysis confirms the dimensional requirement, the core idea is the compatibility condition A (2×3) multiplied by X (3×4) yields B (2×4) with matching inner dimensions and the correct outer dimensions.
If any other options had different dimensions, they would fail the conformability test or yield an incorrect outer dimension for the product AX, given B is 2×4.

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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