Engineering Mathematics Exam > Engineering Mathematics Videos > Inverting 3x3 part 1: Calculating matrix of minors and cofactor matrix
Inverting 3x3 part 1: Calculating matrix of minors and cofactor matrix Video Lecture - Engineering Mathematics
FAQs on Inverting 3x3 part 1: Calculating matrix of minors and cofactor matrix Video Lecture - Engineering Mathematics
| 1. What is meant by the matrix of minors in the context of inverting a 3x3 matrix? |  |
Ans. The matrix of minors is a matrix obtained by replacing each element of a given 3x3 matrix with the determinant of the 2x2 matrix formed by crossing out the row and column of that element.
| 2. How is the matrix of minors used in calculating the cofactor matrix? | |
Ans. The cofactor matrix is obtained by multiplying each element of the matrix of minors by a corresponding sign determined by the position of the element in the original matrix. The sign is positive if the sum of the row and column indices is even, and negative if it is odd.
| 3. What is the purpose of calculating the cofactor matrix in inverting a 3x3 matrix? | |
Ans. The cofactor matrix is used to find the adjugate matrix, which is the transpose of the cofactor matrix. The adjugate matrix is required in the process of calculating the inverse of a 3x3 matrix.
| 4. How does the matrix of minors help in determining if a 3x3 matrix is invertible? | |
Ans. If the determinant of the original matrix is non-zero, then the matrix is invertible. The matrix of minors is used to calculate the determinant, as it involves the determinants of the 2x2 matrices formed from the elements of the original matrix.
| 5. Are there any shortcuts or alternative methods to calculate the matrix of minors and cofactor matrix of a 3x3 matrix? | |
Ans. Yes, there are alternative methods such as using the Laplace expansion or the rule of Sarrus to calculate the determinant of a 3x3 matrix. However, calculating the matrix of minors and the cofactor matrix is a systematic approach that ensures accuracy and can be easily extended to larger matrices.
Text Transcript from Video
I'm now going to do one of
my least favorite things
to do by hand, and that is
to invert a 3 by 3 matrix.
And it can be useful because
you can solve systems that way.
But you'll see it's very
computationally-intensive.
And it is better to
be done by a computer.
And the only thing that's
more painful is doing a 4
by 4 or a 4 by 5 matrix, which
would-- or a 4 by 4 or a 5
by 5 matrix, which
could take all day.
And I'd probably, definitely
make a careless mistake.
But let's just take
it step-by-step.
So the first thing I'm going to
do, this is my 3 by 3 matrix,
is I'm going to construct
a matrix of minors.
So let me construct
here a matrix of minors.
And I'm going to draw this
really big, right over here,
to give ourselves
some real estate.
And the matrix of minors, what
you do is, for each element
in this matrix, you cross
out the corresponding row,
the corresponding column.
And you replace it
with the determinant
of the elements that are left.
So what are left
when you get rid
of this row and this column,
the minor is 1, 1, 4, 5.
So the determinant
of 1, 1, 4, 5.
Let's keep doing that.
This will be
replaced-- well I'll
let you think about that first.
What is this going
to be replaced with?
Well it's going to be replaced--
this row, this column.
The determinant of 2, 1, 3, 5.
Let's keep going.
Let's do this element.
It's going to be replaced with
the determinant of, we get rid
of this row, this
column, 2, 1, 3, 4.
We're a third of the way done,
at least with this first stage
of it.
So for this element
right here, it's
going to be replaced
with its minor.
So we get rid of this
row, this column.
The determinant of
negative 2, 2, 4, 5.
Then we have-- I'm
trying to switch up
the colors reasonably--
this element.
Get rid of the middle
row, middle column.
You're left with the
determinant negative 1, 2, 3, 5.
Now we move on
to-- and I'm really
running out of colors-- this
element right over here,
where its minor is-- we'll get
rid of this row, this column--
negative 1, negative 2, 3, 4.
So let me see that.
Sorry, someone's car alarm
started ringing outside.
So let me make sure I
didn't lose my focus here.
I don't want to make
any careless mistakes.
This row, this column,
negative 1, negative 2, 3, 4.
All right.
Now let's move over here.
Get rid of the first
column, last row.
You have negative 2, 2, 1, 1.
So we have negative 2, 2, 1, 1.
Now let's move to this one.
The middle column, bottom row,
you have negative 1, 2, 2, 1.
So we have negative 1, 2, 2, 1.
And then we are in
the home stretch
for at least the
matrix of minors.
And so we are looking at
this element right over here.
Get rid of the last
column, last row.
You're left with negative
1, negative 2, 2, 1.
So the determinant of
negative 1, negative 2, 2, 1.
And from here we just have
to evaluate each of these
to get the actual
matrix of minors.
This is just a
representation of it.
So let's do that.
So once again, we're
still at the stage
of getting our matrix of minors.
And actually I don't have
to write it as big anymore
because now they're going
to have numeric values.
They're not going to be
these little determinants
of two-by-twos.
So what is the determinant
over here on the top left?
Well it's going to be 1
times 5 minus 1 times 4.
1 times 5 minus 4 times 1.
So it's going to be 5
minus 4, which is 1.
What is the determinant over
here, this blue determinant?
Well it's going to be 2 times
5, which is 10, minus 3 times 1.
So 10 minus 3 is 7.
What is this determinant,
on the top right,
going to evaluate to?
Well you have 2 times 4
is 8, minus 3 times 1.
So it's 8 minus 3, which is 5.
Then we go over here.
What is this determinant
going to evaluate to?
We have negative 2 times 5 is
negative 10, minus 4 times 2.
So it's negative 10 minus
8, which is negative 18.
Then we have negative 1
times 5, which is negative 5,
minus 3 times 2.
So it's negative 5 minus
6, which is negative 11.
I want to do that in
white, negative 11.
What's this determinant
going to evaluate to?
We have negative 1 times 4,
negative 4, minus negative 6.
So that's negative 4 plus
6, which is positive 2.
I want to do that in that--
so that is positive 2.
We have three left.
What does this evaluate to?
Negative 2 times 1 is
negative 2, minus 1 times 2.
So it's negative 2 minus
2 gets us to negative 4.
Home stretch now.
Negative 1 times negative 1 is
negative 1, minus 2 times 2.
So it's negative 1 minus
4, which is negative 5.
And then finally we
have negative 1 times 1
is negative 1, minus
2 times negative 2,
so minus negative 4.
So it's negative 1
minus negative 4.
That's the same
thing as adding a 4.
So it's negative 1 plus 4.
So it's going to
be a positive 3.
So this right over here is
our true matrix of minors.
And from there we
can get our cofactor.
We can get our
cofactor matrix just
by remembering a
checkerboard pattern.
So a checkerboard
pattern tells us
positive, negative, positive,
negative, positive, negative,
positive, negative, positive.
It's a little
self-explanatory why
that's called a checkerboard.
So if we sign this matrix
of minors in this pattern,
then we get our cofactor matrix.
So let's set up our cofactor
matrix right over here.
So this is our cofactor.
A lot of terminology,
but hopefully it's
making a little bit of sense.
Our cofactor matrix.
So we just have to apply
these signs to these values,
to the matrix of minors.
So 1 is now going to have
applied a positive sign to it.
So it's still just going
to be a positive 1.
You're going to have
7, but it's going
to have a negative
sign applied to it.
So it's going to
be a negative 7.
You have a 5, positive 5.
The 5 is already positive.
You multiply it
times a positive,
it's going to be a positive.
You have negative
18, but then we
have to multiply that
times a negative.
So you get positive 18.
You have negative 11, multiply
that times a positive 1.
And so you still
have negative 11.
You have positive 2,
multiply by a negative 1.
You're going to have negative 2.
Then you have negative 4,
multiply by a positive 1.
You still have negative 4.
And then you have negative 5.
What is this going to be
in the cofactor matrix?
Well you take a negative 5,
multiply by a negative 1.
You have a positive 5.
And then finally, you
have a positive 3,
multiply by a positive 1.
You're still going
to have a positive 3.
So we've gone pretty
far in our journey,
this very
computationally-intensive
journey-- one that
I don't necessarily
enjoy doing-- of finding
our inverse by getting
to our cofactor matrix.
Now we just have to
take this determinant,
multiply this times 1 over the
determinant and we're there.
We've figured out the
inverse of matrix C.
my least favorite things
to do by hand, and that is
to invert a 3 by 3 matrix.
And it can be useful because
you can solve systems that way.
But you'll see it's very
computationally-intensive.
And it is better to
be done by a computer.
And the only thing that's
more painful is doing a 4
by 4 or a 4 by 5 matrix, which
would-- or a 4 by 4 or a 5
by 5 matrix, which
could take all day.
And I'd probably, definitely
make a careless mistake.
But let's just take
it step-by-step.
So the first thing I'm going to
do, this is my 3 by 3 matrix,
is I'm going to construct
a matrix of minors.
So let me construct
here a matrix of minors.
And I'm going to draw this
really big, right over here,
to give ourselves
some real estate.
And the matrix of minors, what
you do is, for each element
in this matrix, you cross
out the corresponding row,
the corresponding column.
And you replace it
with the determinant
of the elements that are left.
So what are left
when you get rid
of this row and this column,
the minor is 1, 1, 4, 5.
So the determinant
of 1, 1, 4, 5.
Let's keep doing that.
This will be
replaced-- well I'll
let you think about that first.
What is this going
to be replaced with?
Well it's going to be replaced--
this row, this column.
The determinant of 2, 1, 3, 5.
Let's keep going.
Let's do this element.
It's going to be replaced with
the determinant of, we get rid
of this row, this
column, 2, 1, 3, 4.
We're a third of the way done,
at least with this first stage
of it.
So for this element
right here, it's
going to be replaced
with its minor.
So we get rid of this
row, this column.
The determinant of
negative 2, 2, 4, 5.
Then we have-- I'm
trying to switch up
the colors reasonably--
this element.
Get rid of the middle
row, middle column.
You're left with the
determinant negative 1, 2, 3, 5.
Now we move on
to-- and I'm really
running out of colors-- this
element right over here,
where its minor is-- we'll get
rid of this row, this column--
negative 1, negative 2, 3, 4.
So let me see that.
Sorry, someone's car alarm
started ringing outside.
So let me make sure I
didn't lose my focus here.
I don't want to make
any careless mistakes.
This row, this column,
negative 1, negative 2, 3, 4.
All right.
Now let's move over here.
Get rid of the first
column, last row.
You have negative 2, 2, 1, 1.
So we have negative 2, 2, 1, 1.
Now let's move to this one.
The middle column, bottom row,
you have negative 1, 2, 2, 1.
So we have negative 1, 2, 2, 1.
And then we are in
the home stretch
for at least the
matrix of minors.
And so we are looking at
this element right over here.
Get rid of the last
column, last row.
You're left with negative
1, negative 2, 2, 1.
So the determinant of
negative 1, negative 2, 2, 1.
And from here we just have
to evaluate each of these
to get the actual
matrix of minors.
This is just a
representation of it.
So let's do that.
So once again, we're
still at the stage
of getting our matrix of minors.
And actually I don't have
to write it as big anymore
because now they're going
to have numeric values.
They're not going to be
these little determinants
of two-by-twos.
So what is the determinant
over here on the top left?
Well it's going to be 1
times 5 minus 1 times 4.
1 times 5 minus 4 times 1.
So it's going to be 5
minus 4, which is 1.
What is the determinant over
here, this blue determinant?
Well it's going to be 2 times
5, which is 10, minus 3 times 1.
So 10 minus 3 is 7.
What is this determinant,
on the top right,
going to evaluate to?
Well you have 2 times 4
is 8, minus 3 times 1.
So it's 8 minus 3, which is 5.
Then we go over here.
What is this determinant
going to evaluate to?
We have negative 2 times 5 is
negative 10, minus 4 times 2.
So it's negative 10 minus
8, which is negative 18.
Then we have negative 1
times 5, which is negative 5,
minus 3 times 2.
So it's negative 5 minus
6, which is negative 11.
I want to do that in
white, negative 11.
What's this determinant
going to evaluate to?
We have negative 1 times 4,
negative 4, minus negative 6.
So that's negative 4 plus
6, which is positive 2.
I want to do that in that--
so that is positive 2.
We have three left.
What does this evaluate to?
Negative 2 times 1 is
negative 2, minus 1 times 2.
So it's negative 2 minus
2 gets us to negative 4.
Home stretch now.
Negative 1 times negative 1 is
negative 1, minus 2 times 2.
So it's negative 1 minus
4, which is negative 5.
And then finally we
have negative 1 times 1
is negative 1, minus
2 times negative 2,
so minus negative 4.
So it's negative 1
minus negative 4.
That's the same
thing as adding a 4.
So it's negative 1 plus 4.
So it's going to
be a positive 3.
So this right over here is
our true matrix of minors.
And from there we
can get our cofactor.
We can get our
cofactor matrix just
by remembering a
checkerboard pattern.
So a checkerboard
pattern tells us
positive, negative, positive,
negative, positive, negative,
positive, negative, positive.
It's a little
self-explanatory why
that's called a checkerboard.
So if we sign this matrix
of minors in this pattern,
then we get our cofactor matrix.
So let's set up our cofactor
matrix right over here.
So this is our cofactor.
A lot of terminology,
but hopefully it's
making a little bit of sense.
Our cofactor matrix.
So we just have to apply
these signs to these values,
to the matrix of minors.
So 1 is now going to have
applied a positive sign to it.
So it's still just going
to be a positive 1.
You're going to have
7, but it's going
to have a negative
sign applied to it.
So it's going to
be a negative 7.
You have a 5, positive 5.
The 5 is already positive.
You multiply it
times a positive,
it's going to be a positive.
You have negative
18, but then we
have to multiply that
times a negative.
So you get positive 18.
You have negative 11, multiply
that times a positive 1.
And so you still
have negative 11.
You have positive 2,
multiply by a negative 1.
You're going to have negative 2.
Then you have negative 4,
multiply by a positive 1.
You still have negative 4.
And then you have negative 5.
What is this going to be
in the cofactor matrix?
Well you take a negative 5,
multiply by a negative 1.
You have a positive 5.
And then finally, you
have a positive 3,
multiply by a positive 1.
You're still going
to have a positive 3.
So we've gone pretty
far in our journey,
this very
computationally-intensive
journey-- one that
I don't necessarily
enjoy doing-- of finding
our inverse by getting
to our cofactor matrix.
Now we just have to
take this determinant,
multiply this times 1 over the
determinant and we're there.
We've figured out the
inverse of matrix C.
Related Exams
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Nov 18, 2024 Last updated |
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