Yes there *can be* many answers to `N` in Part 2 but one thing that wasn't stated is that Part 2 only carries 2 marks in the actual question.

So in the interest of grabbing this puny 2 marks in the quickest of time under stressful exam conditions, it's not really advisable to go through that lengthy 長氣workings of yours (even though your final answer is correct).

A simple recall of the Identity Matrix (which is in this case) that satisfies the conditions of `N` ought to do the trick 😉

1. AB =

A

(AB)

OK to make things clearer, a matrix is **invertible** if it is a square matrix i.e. `n` by `n` AND if its determinant ≠ 0.

So A^{-1} does not exist since A is not even a square matrix to begin with.

For (AB)^{-1}, MAKE SURE you do a quick calculation of its determinant i.e.

just to check that it's not zero before stating that the inverse exists!

P.S. your ^{-1} should appear properly now 😉

2. Let N = .

MN=

MN=

NM=

NM=

By equality of matrices,

4a+5b=4a+2c -->5b=2c

4c+5d=5a+3c --> c+5d=5a --(1)

2a+3b=4b+2d --> 2a=b+2d--(2)

2c+3d=5b+3d --> 2c=5b

.'. b:c = 2:5

Let's take b=2k and c=5k where k is any number.

(1): 5k+5d=5a

k+d=a

a-d=k=b/2=c/5

(2):2a=2k+2d

a-d=k=b/2=c/5

So we need to find values such that a-d=b/2=c/5.

If we take b=2 and c=5, then a-d=1

a and d **can be** 1 and 0 respectively.

Checking...MN=; NM= .'. N **can be**.

Of course there are many other solutions to this question.你认为矩阵(matrices)麻烦吗？之前的ratios问题(https://www.exampaper.com.sg/blog/questions/e-maths/similarity-ratios-fetish)中，其实ratios叫「比」。

Please see Miss Loi's comment #6 below.

Why hide part of your comments? It's so interesting for 新加坡個小朋友 to know that Matrices=「矩阵」 and Ratios=「比」!

A matrix whose determinant is zero is

singularand hasno inverse.On the other hand, a matrix whose determinant is not zero is

non-singularand has aninverse.

But the tricky part here is that A is a non-square matrix. So can you even calculate its determinant? So can A ever be non-singular? 🙂

]]>Can i just say .... "A is a non-singular matrix, hence there is an inverse matrix for A"? ]]>