Given the matrices [math: A=[25−34−10]] A= \left[\begin{array}{cc} 2&5 \ -3&4 \ -1&0 \end{array} ight] and [math: B=[4−20−113]] B= \left[\begin{array}{cc} 4&-2 \ 0&-1 \ 1&3 \end{array} ight] , if [math: AX=B] the order of matrix X is

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Given the matrices [math: A=[25−34−10]] A= \left[\begin{array}{cc} 2&5 \ -3&4 \ -1&0 \end{array}ight] and [math: B=[4−20−113]] B= \left[\begin{array}{cc} 4&-2 \ 0&-1 \ 1&3 \end{array}ight] , if [math: AX=B] the order of matrix X is

BlackTom AI · Accepted Answer

First, note the given matrices' dimensions to understand how matrix multiplication works. A is described as having 3 rows and 2 columns, so A is a 3×2 matrix. The product AX must equal B, and B is given as 3×2, so B is a 3×2 matrix. For the product A×X to be defined, X must have a number of rows equal to the number of columns of A, i.e., X must be 2×p for some p.

Since the resulting matrix AX has the same number of rows as A (which is 3), the number of rows of AX is 3, agreeing with B's row count. The column count of AX is p, and this must match B's column count, which is 2. Therefore p = 2.

Putting this together, X must be a 2×2 matrix so that A (3×2) times X (2×2) yields a product of size 3×2, matching B.

Now let's evaluate the given options:

- Option a: 3×2 would require X to have 3 rows, but X must have 2 rows to align with A's 2 columns, so a 3×2 X is not dimensionally compatible with A for multiplication.
- Option b: 2×3 would give A (3×2) times X (2×3) resulting in a 3×3 matrix, which does not match B's 3×2 size.
- Option c: 3×3 would yield a 3×3 product, again incompatible with B being 3×2.
- Option d: 2×2 fits perfectly, since A (3×2) times X (2×2) produces a 3×2 matrix, matching B.

Thus, the only dimensionally consistent choice is 2×2 for X.

Given the matrices [math: A=[25−34−10]] A= \left[\begin{array}{cc} 2&5 \\ -3&4 \\ -1&0 \end{array}\right] and [math: B=[4−20−113]] B= \left[\begin{array}{cc} 4&-2 \\ 0&-1 \\ 1&3 \end{array}\right] , if [math: AX=B] the order of matrix X is

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