Need math?
Dude, no path.
Fills us with wrath
Swells us with wreath
Oh! Not to grave
But to grow.
Math, math, math,
Mother of truth
Author of faith
Dude, it's the path.
In depth of doubt
Math keeps us straight.
Work math at home
Strike with warm chime.
Thursday, December 03, 2015
Thursday, November 12, 2015
Determining the Determinant of a 3x3 matrix
Introduction
Let us consider the
2x2 matrix given on the left. Its determinant as per the rule given above is ADBC.
However, look at the
diagram given right. It can be explained as follows.
Rotation Method
Rotate left the
second row to one position and then multiply the entries in the respective
columns. Subtract the second column product from that of the first.
This process can be
extended into higher orders also. This
is done by carefully completing all the rotations depending on the number of
rows. Let us see the case of a 3x3 matrix.
At first, rotate left
the second row to one position and third row to two positions. The matrix thus
obtained is given on the right.
Thus we get A= (A1.B2.C3)+(A2.B3.C1)+(A3.B1.C2).
Now we rotate left
the second row to one more position and the third row to two more positions.
The matrix thus obtained is as follows.
Multiply the values
in each column separately and then add the three products. Thus we get B=(A1.B3.C2)+(A2.B1.C3)+(A3.B2.C1).
The determinant of
the 3x3 matrix is then
AB = (A1.B2.C3+A2.B3.C1+A3.B1.C2)( A1.B3.C2+A2.B1.C3+A3.B2.C1).
Cylindrical
Rotation Method
This process is
better understood if we can represent the matrix cylindrically. Consider a
cylinder that has three horizontal sections which can be rotated freely with
respect to a central vertical axis. Entries in the matrix are given on the
exterior of the cylinder.
After the first set
of two types of left rotations, the cylinder looks like what is on the left.
Multiplying the
entries in the columns separately we get
A=(A1.B2.C3)+(A2.B3.C1)+(A3.B1.C2).
After the second set
of two types of left rotations, the cylinder looks like what is on the right.
B = (A1.B3.C2)+(A2.B1.C3)+(A3.B2.C1).
The determinant of
the 3x3 matrix is then
AB = (A1.B2.C3+A2.B3.C1+A3.B1.C2)( A1.B3.C2+A2.B1.C3+A3.B2.C1).
Palm Method
This can be
visualized in yet another way also.
Let us use the three
central fingers on the left palm to represent a 3x3 matrix.
Instead of the first
set of rotations, multiply the entries from left as indicated by the dark
lines, starting from the diagonal to get the value A.
Conclusion
An advantage of the
above mentioned process is the elimination of the repeated use of plus(+) and minus(–)
which is sometimes disturbing for beginners and nonMathematics students. Can this method be extended to higher order matrices?
Monday, September 07, 2015
Before I became a Teacher
Before I became a Teacher never I
knew
That sleeping in the class disturbs
The teaching of the teacher.
That talking in the class makes
The teacher shouting in the class.
That not scribbling in the class augments
The degrading of the student.
That correcting the answer scripts
is tougher
Than attempting all the questions.
That questioning the answers is harder
Than answering the questions.
That coming late for the class is
worse
Than not attending the class.
That absenting in the class is worse
Than not joining for the course.
That photocopying is the greatest
insult
That a teacher can ever bear.
That helping a fellow student
learn is
The second greatest insult to a
teacher.
That teaching is the noblest profession
When all your students become teachers.
Saturday, September 05, 2015
The Teacher I adore and Fear...
First impression is the best
But he proved me wrong, my trust
By being different, still a good teacher
Never roamed behind us like a preacher
Fills my eyes with few tears
As I give my ears when he is sad and cares
And he sends deep in me shivers
When he is angry and brings in my fears
Still he tries his best to put me in front
Although we never succeed, he didn't grit
Waiting, giving us our needed space
Putting us equally with the race
Also makes me admire his knowledge
Which through his words clearly acknowledges
Am sure greater heights are awaiting
To celebrate his humble lovely teaching
He is my teacher whom I adore
But fears put me back far from his doors
Doors of wisdom that is open to all
Spread even if no inner voice gives a call.
On the height of my selfdoubt, given to me by one of my students unexpectedly on the Teacher's Day of 2015.
But he proved me wrong, my trust
By being different, still a good teacher
Never roamed behind us like a preacher
Fills my eyes with few tears
As I give my ears when he is sad and cares
And he sends deep in me shivers
When he is angry and brings in my fears
Still he tries his best to put me in front
Although we never succeed, he didn't grit
Waiting, giving us our needed space
Putting us equally with the race
Also makes me admire his knowledge
Which through his words clearly acknowledges
Am sure greater heights are awaiting
To celebrate his humble lovely teaching
He is my teacher whom I adore
But fears put me back far from his doors
Doors of wisdom that is open to all
Spread even if no inner voice gives a call.
On the height of my selfdoubt, given to me by one of my students unexpectedly on the Teacher's Day of 2015.
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