Scalar Multiplication in Determinant and Matrices

The study of determinants and matrices can be confusing at time. They look almost the same, and are closely related. As such, some mathematical operations on them are similar.

But, there are differ somewhat and have to be taken care of.

Below is one such mistake that is often made by learning math students.



The upper row involves matrices. The scalar multiplication of it by "a" affects ALL the elements within the matrix.

For the lower row, the scalar "a" multiplication affects only one column (or one row) of the determinant.

Just note this different and commit it to memory. This is good for you and can save you a lot of further troubles with matrix and determinant.

So, just simply identifying the differences between matrices and determinant, in term of their operations, you will be enjoying them for a long time.

With less mistake made, you will be more gear towards doing a better job out of maths. This is human nature.

Cheers! and Happy mathematics!
:D

How To Transpose A Matrix

Transposing a matrix is a term for a process that "swap" the row of a matrix with its column.

All the elements within the matrix remain except for their placement.

NOTE: Matrix transpose is important and a key step in getting the Inverse matrix which is required in the solving of simultaneous equations.

The procedure to transpose is simple, but saying that, caution has to be taken of which direction to "rotate". Look carefully at the coloured elements to see the direction (below diagram).


Here you see that the symbol to represent "Transpose" s the "T" besides the original matrix. The elements remain intact and not shifted yet.

Only upon transposing, the "T" disappeared.

Now, see how the transposed elements are done.
The first row is changed to the first column, with the element "a" being the pivot point.

Common mistake:
The pivot point (element) is taken wrongly. Normally element "c" is the target which is incorrect!

Stay focus when doing this transpose. You will then rid yourself of less trouble.

Cheers! Remember, maths is interesting! (If you are doing the correct thing.)
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How to do the 3 x 3 order determinant using Rule of Sarrus

It is very often that we come across the word determinant in matrices. Yes, they are related.

Matrix consists a set of numbers. That's it.
Determinant is a numerical value obtained throught the numbers within the matrix.

For a 3 x 3 matrix, how then do we extract the determinant?

One of the easiest method to obtain this number is throught the use of Rule of Sarrus.

Below I show the steps.


Determinant A = [(aei) + (bfg) + (cdh) ] - [ (gec) + (hfa) + (idb) ].

=====================================================>>>

The important step lies with getting the first left 2 columns out to the right of the determinant.

After which, we do a downwards grouping and addition.
This is followed by upwards grouping with addition again.

Note, the last step is to SUBTRACT the two groups obtained above.

(By grouping, I mean to MULTIPLY the individual elements).

Common mistakes:
1) The grouping is done by "adding" instead of multiplying the elements.

2) The slanted operations (the 3 element-multiplication) are done by "adding" instead of multiplying the 3 elements.

3) The two big groups are "added" instead of subtracted".

Just bear in mind the 3 common mistakes above and you are on the way to a happy matrix student...

Cheers!

Multiplication of Matrices

As do normal number multiplication, matrices themselves, also have their set of multiplication.

They are however done in another interesting way.

This is so since a matrix consists of a set of numbers within it.

Therefore multiplying matrices are not so straight forward as normal numbers.

But do rest assure that there is nothing new, it is still simple multiplication ( and addition / subtraction).

Maths is simple and interesting, remember? Read on for the steps and caution points.


Below I show multiplication using 2 matrices of the 2 x 2 order.

Do note that I have presented the steps first, leaving out the technical computation later on in this post (below the diagram).




See below for the steps:

• element "K" is obtained from the first row of left matrix and the first column of the next matrix,
• element "L" is obtained using first row of left matrix and SECOND column f next matrix.
• element "M" is obtained using SECOND row of left matrix and first column of next matrix.
• element "P" is using second row and column of the respective matrices.

These sequence is of utmost importance. Why?

By not following the sequence to obtain the final elements, the numbers will be simply messed up! This is a very common mistake among students of matrices.

It is mental discipline to systematically derive the elements in the appropriate positions.

Next, let us go into getting the actual number for the final elements. Here we go...

Elements "K" = (4 x a) + (2 x c)

Element "L" = (4 x b) + (2 x d)

Element "M" = (1 x a) + (5 x c)

Element "N" = (1 x b) + (5 x d) ===> There is it, you have gotten all the elements!

There is nothing difficult, right? Matrix study is one of the simplest topics in maths.

Have fun!

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Matrices | What is Minor?

Before you read this, say "maths is interesting!". Mental trick....
Now you may start.

In matrices, there is an important basic term residing in it.

It is the term called "minor" and is obtained through the determinant of the matrix.

For 2nd order matrix (or determinant), there will be 4 minors since there are 4 elements within the determinant.
For 3rd order form, there will inevitably, be 9 minors.


To have an idea what "order of matrix" is about, go to this link here.


How to compute this minor?

See below...

The method is simple, right?
It involves the correct cancellation of the appropriate row and column.

The leftover elements will be used to determine the value.

To enhance understanding, let's do another example.Find the minor of element 2.
The leftover is the elements
4 6
7 9
The determinant value of the leftover is then (4)(9) - (7)(6) = 36 - 42 = -6.
This is the minor for element "2".

Common mistake made:
The minor is always taken as the actual element in question.

Example of mistakes:
To find the minor of element "2" in the above determinant
==> Wrong answer = "2", the original element used to get the leftover after cancellation.

To find the minor of element "1" in the above determinant
==> Wrong answer = "1", the original element used to get the leftover after cancellation.

NOTE:
Minor and Co-factor are closely related. They are needed to compute Adjoint and Inverse matrix.

You have to master these concepts to be able to handle matrices.

Tough, right? If your answer is YES, you are wrong!
Look and review the above methods again, and you will be sure that they are simply +, _, x only.
Isn't this what you have been learning since lower elementary?

:-)

How To Get Determinant Value

In the study of matrices, you will come across the computation of determinant and their values.
These determinants come in various order.
There are the 2nd order, 3rd order, and so on.

What is this order?

Order is the number of rows and column within the determinant.
Thus, 2nd order determinant has 2 rows and 2 columns.
3rd order has 3 rows and 3 columns.


To calculate the values of these determinants, see below for the methods.


The 2nd order is shown in the first upper line.
Following that is the 3rd order computation, using the special Rule of Sarrus (the simpler form of 3rd order calculation).

This value computed comes in useful to determine other values.
One of the examples is the solution of simultaneous equations and their unknowns.
The technique presented here is thus, basic knowledge in matrices learning.

Adjoint and Inverse Matrix } Their relationship

There is a common misunderstanding among math learners over the common term in matrices.
The two terms are the Adjoint and Inverse matrix.

The operation to get them is somewhat always confused and "mixed up".

Here to clear and simplify their differences:
- Adjoint matrix
It is obtained through the co-factors of each elements within the matrix. After which the matrix is transposed. It stops there.

- Inverse matrix
The adjoint matrix forms part of this inverse matrix!
Inverse matrix needs a further computation of the determinant also.
The final outcome is the adjoint matrix divided by this determinant value.

Therefore in summary, inverse matrix is formed out of the adjoint matrix.
They are husband and wife, with the kids, the minor and co-factors in them.

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Inverse Matrix | What is it used for?

One of the applications of inverse matrix is in solving simultaneous equations.

If you are good with algebra, you will discover that this inverse matrix way of handling the solution of simultaneous equations is similar, except that they are done as a group, collectively.
The answers to the unknown variables are obtained at one go with this Inverse matrix method.

However, what do you need to know in order to use this Inverse matrix solving?

You have to understand:
1) Convertion of simultaneous equations into a set of matrices
2) Determinant and technique to get its numerical value
3) Minor of the individual elements within the matrix
4) Co-factor of this determinant formed with this set of matrices
5) Transpose of matrix
6) Adjoint matrix obtained with the co-factors and transposed matrix
7) Formula to relate determinant with the adjoint matrix ==> Inverse matrix
8) Matrices multiplication

The list looks amazingly long for matrix novice, but, DO NOT FEAR!

Why?
Matrices consist of numbers only, and simple mathematical operations, nothing abstract.
(The details are not presented here for fear that you will leave this site.)

Slowly research into the above terms and see for yourself that they are "friends" and not "foes".

Happy start to matrices and its application.
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Differences between Determinants and Matrices

Determinants and matrices, they look alike. Their similarities caught many unaware and results in "excitements" and much interests.

Both contain numbers within. But ......

- determinants are bounded by two straights lines whereas matrices are by square braces

- determinant resulted in a single numerical value, whereas matrices are sets of numbers grouped within the braces

- determinant can be extracted from matrix, but not the other way round

- there are inverse matrix but not inverse determinant

- a scalar multiplier affects only a single row or single column of a determinant, but affects all the numbers within a matrix

The differences are aplenty. But those listed above are the least any maths students or person should know.

Why create such topics to excite maths learners?
Answer: So that maths will be made interesting and challenging (... the informal reply)

With open mindedness, maths is a very fantastic subject to learn and train for. It carries many hard works and beautiful thinking. Thoughts that are simple yet abstract at times.

Any more fun? ..... You are the answer!

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Cramer and Sarrus | How are they related to maths

Cramer and Sarrus?

Who are they?

In which area of maths are they involved?

For those of you who is doing matrices, you will be familiar with their rules, I suppose.

They are famous for creating techniques used in the solution of simultaneous equations.

How so?

Cramer's Rule is a method of solving unknowns in simultaneous equations using division of various determinants.

Rule of Sarrus is a simple way of determining the numerical value of 3rd order determinant.

The two rules, therefore, complement each other to form an easy way to solve simultaneous equations.

However do note that students of these topics use to mix up the 2 rules.

When question states the use of Cramer's and Sarrus rule, they tend to get confuse. THe common notion is that Rule of Sarrus is another method to solve simultaneous equations which is not so.

It is only a technique to evaluate the determinant and get its value. It cannot solve the unknowns in the set of equations.

This is a good problem to get confuse. Why?

Getting confuse means a thinking brain and learning starts to take place. When there is no confusion, there can be two happenings.
- blank or "completely do not understand"
- "completely understood without questions"

Which is which does not matter. What is important is not to disappoint Cramer and Sarrus!
Work hard folks!

Cheers!
:-)