In order to put Miss Loi’s new found freedom into good use, it’s about time the Matrix is loaded onto this website … hohoho.

Do take particular note on the second question of Part 1. Recently Miss Loi’s been seeing that more and more school papers are testing students’ understanding of concepts like these (which is the right thing to do).

From experience, many students don’t pay enough attention to mundane little things like the conditions for inverses to exist. Even Miss Loi is sometimes guilty of that in her uni days :P.

Given that A = and B = , find AB. State, with reason, whether A^{-1} and (AB)^{-1} exist.

Given that M = , find the matrix N such that MN = NM.

1. AB = A^{-1} does not exist because it isn't a square matrix. (AB)^{-1} exists because it's a square matrix.

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.

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 😉

* Please refer to the relevant Ten-Year Series for the questions of these suggested solutions.

About Miss Loi

Miss Loi is a full-time private tutor in Singapore specializing in O-Level Maths tuition. Her life’s calling is to eradicate the terrifying LMBFH Syndrome off the face of this planet. For over years she has been a savior to countless students … [read more]

Gratitudes

If it wasn’t because of your teachings and enlightenment, I wouldn’t be here today, I wouldn’t be taking A-Maths, and Maths does not seem THAT difficult anymore … [read more]

## 10 Comments

曜

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How to prove for part 1 ar?

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

曜

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123, Miss Loi shall reproduce this from an obscure corner of your text-book

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? 🙂

曜

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1. AB =

A

^{-1}does not exist because it isn't a square matrix.(AB)

^{-1}exists because it's a square matrix.OK to make things clearer, a matrix is

invertibleif it is a square matrix i.e.nbynAND 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 be1 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」!

曜

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I wonder why my -1 does not go superscript.

曜

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Thanks!

曜

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Wait

Li-saMiss Loi hasn't dismissed you yet!Yes there

can bemany answers toNin 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

Nought to do the trick 😉曜

日

Maybe I should tell you that I have hidden some information in the source code.

曜

日

*無奈*

曜

日

I also tried the "This is A-Maths not Physics".

曜

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Oh, and "Thanks!" is not equal to fleeeeeeing and somehow I wonder how you linked it to unintended dismissal. It is what it it is.