What is that?

You are finally awake my friend!

Oh no, you have lost your memory after fighting with the monster Lanier. 

Well, I am Matt, your best friend, and YOU are the inheritor of our kingdom Matrixland. 

Hurry up! Lanier has caught away our villagers. We need to fight Lanier and save our villagers!

Oh, my majesty, my name is Alpha. My friend Beta had been taken away. Please save him!

To help you, let me introduce you a powerful weapon: the matrix.

What is that?

It is just a group of numbers. However, it is arranged in a special way.

Look, we can think of it as a cubby hole, and each hole stores a number. The vertical line is called the column, and the horizontal line is called the row, while the dimension is the number of rows and columns of the matrix.

Look, here is a 3 x 3 matrix on the left, and a 2 x 3 matrix on the right.

Interesting! Thanks Alpha, I'll be on my way.

A =

To break,

press a

Oh no! The stone has blocked our way to the village!

But look, there is a matrix problem on the stone. I think we have to press the right number on the matrix to break the stone.

To break,

press a

23

A =

correct !

Since the dimension of a matrix represents the number of rows and columns, does this mean that the element of the matrix we need to press belongs in the 2nd row and 3rd column? 

Let's try and press 4.

Hi, I'm Eddy. Welcome to Math Kingdom!

Don't be glad so soon, because the monsters that caught our villagers are getting harder and harder to defeat.

In order to defeat them, you need to learn the skill of combining matrices.

We did it! We are through to go to the village.

Oh, look! A villager!

Wow! How do we do that?

I will first teach you the skill to perform matrix addition. We can add matrices together, just like the addition of numbers like 1 + 1 = 2.

In matrix addition, we can add matrices with the same dimension by adding the entries of each position together, and it becomes a new matrix!

So for example,

For the number in the first row and first column, we have 1 in the first matrix and 5 in the second matrix. Therefore, 1 + 5 gives us 6. Same goes for the others.

Interesting...

Here come two small monsters.

Look! On that board are four matrices. If we can make two matrices by adding the matrices on that board, we can make a weapon powerful enough to defeat them.

Let us think... As Eddy said earlier, we can only add matrices if they have the same dimension. The top left and bottom right matrices are both 2 by 2, while the top right and bottom left matrices are both 2 by 3. Can you help me solve this matrix addition?

Thanks! We now have the two matrices we need!

Now let's try and defeat those monsters.

The monsters have been scared away by these two matrices. However...

Oh no, before running away, the monsters threw four matrices in our direction! We can’t handle it at the same time!

Don't worry my Friend! I am here to help you.

We can do subtraction! The rule is similar to that of addition.

See, we can subtract the red matrix with the yellow matrix, by subtracting the entries of each position together.

Note since we have 12 - 5 = 7 for the first entry, 

and 21 - 7 = 14 for the second entry, we obtain a new matrix

Oh, there you are, Matt! Wow, let me have a try. I have the green and the blue matrices flying to me. If I do the subtraction of each entry, I get

Nice, we have solved two subtraction problems. Crisis averted, good job! But, who is crying...?

Anyone, help! I am Omega. I have been trapped by this matrix lock. It can only be opened by the scalar multiplication of the matrix. 

It sounds difficult but don't be scared. You just need to know that any matrix can be multiplied by a number, called the scalar.

For instance, if we have      , and we want to multiply it by the scalar 2, we have to multiply each element of the matrix by 2, so that the first entry will be 2x2=4, and the second entry will be 2x3=6.

Look, we now have

Got it! Let me help you. The lock says                             and -2.

Wow, this is a tricky matrix, but we can solve it!

If we multiply                      by -2, we need to

 multiply each entry by -2, so we get

The password must be it! Let me enter the password for you, Omega.

Great! The lock is opened!