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

Question

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

BlackTom AI · Accepted Answer

We start by restating the given problem and options to anchor the discussion.
Question: Given the matrices A and B, determine the order (dimensions) of the product ABAB.
Options: 
c. 3x3

Next, we analyze the dimensions of A and B individually. Matrix A is given as a 3-by-2 matrix (3 rows, 2 columns): A is 3x2. Matrix B is given as a 2-by-3 matrix (2 rows, 3 columns): B is 2x3.
Now, consider the product AB. The inner dimensions must match: A is 3x2 and B is 2x3, so the product AB is defined and has dimensions 3x3 (the outer dimensions: 3 from A, 3 from B).
With AB determined to be 3x3, we can form the product ABAB as (AB)(AB). For the second multiplication to be defined, AB must be a square matrix (which it is, 3x3), allowing us to multiply AB by AB.
When multiplying two 3x3 matrices, the result is again a 3x3 matrix. Therefore, the overall product ABAB has dimensions 3x3.
In this context, option c. 3x3 correctly states the resulting dimensions.

Why this option is correct relative to the other possibilities (if they existed): If you considered dimensions that do not align, such as a mismatch in inner dimensions (e.g., trying to multiply a 3x2 by a 3x2 directly), the product would be undefined. However, since AB is defined and yields 3x3, the subsequent multiplication ABAB is also defined and yields 3x3. The stated option matches the actual dimensional outcome of the sequence of multiplications.

Alternative viewpoints and common misconceptions:
- A common pitfall is to incorrectly multiply A and B without checking inner dimensions, which can lead to assuming wrong shapes. Here, AB is valid because 2 (A's columns) matches 2 (B's rows).
- Another mistake is to think ABAB would have non-square or different dimensions; since AB is 3x3, ABAB is also 3x3, not something like 3x2 or 2x3.
- Finally, some students might misinterpret the order or forget to apply the outer-dimension rule when multiplying sequentially; remembering that the result's rows come from the leftmost matrix and the result's columns come from the rightmost matrix helps avoid this.

MCD1550 / MCD2140 - T3 - 2024 Test 3

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

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