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

Question

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

查看解析

登录即可查看完整答案

类似问题

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

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

If A is 5×75\times7 then A⊤A^\top is [Fill in the blank], ×\times [Fill in the blank], .

The order of the matrix M=[2−300] M = \left[\begin{array}{cccc} 2&-3&0&0 \end{array}\right] is

Given the matrices A=[−103−246] A = \left[\begin{array}{cc} -1&0 \\ 3&-2 \\ 4&6 \end{array}\right] and B=[5−201−15] B = \left[\begin{array}{ccc} 5&-2&0 \\ 1&-1&5 \end{array}\right] , the order of the product matrix AB AB is

The order of the matrix [math: M=[2−300]] M = \left[\begin{array}{cccc} 2&-3&0&0 \end{array}\right] is

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单