JEE Exam > JEE Videos > Mathematics (Maths) Class 12 > Examples : Finding Inverse of 3x3 Matrices
Examples : Finding Inverse of 3x3 Matrices Video Lecture | Mathematics (Maths) Class 12 - JEE
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FAQs on Examples : Finding Inverse of 3x3 Matrices Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. How do I find the inverse of a 3x3 matrix? |  |
Ans. To find the inverse of a 3x3 matrix, you need to follow these steps:
1. Calculate the determinant of the matrix.
2. If the determinant is non-zero, proceed to the next step. Otherwise, the matrix does not have an inverse.
3. Find the matrix of minors by replacing each element with the determinant of the 2x2 matrix formed by the remaining elements.
4. Create the matrix of cofactors by multiplying the matrix of minors by the appropriate sign (+ or -) based on the element's position.
5. Transpose the matrix of cofactors to get the adjugate matrix.
6. Finally, divide the adjugate matrix by the determinant to obtain the inverse of the 3x3 matrix.
| 2. What is the determinant of a 3x3 matrix? | |
Ans. The determinant of a 3x3 matrix can be calculated using the following formula:
Determinant = (a * e * i) + (b * f * g) + (c * d * h) - (g * e * c) - (h * f * a) - (i * d * b)
Here, a, b, c, d, e, f, g, h, and i represent the elements of the 3x3 matrix.
| 3. Can a 3x3 matrix have a zero determinant? | |
Ans. No, a 3x3 matrix cannot have a zero determinant if it is invertible. If the determinant of a 3x3 matrix is zero, then it is singular and does not have an inverse.
| 4. What happens if a 3x3 matrix does not have an inverse? | |
Ans. If a 3x3 matrix does not have an inverse, it is called a singular matrix. In this case, the matrix cannot be inverted, and its determinant is zero. Singular matrices often represent systems of linear equations that have no unique solution or have infinite solutions.
| 5. Is finding the inverse of a 3x3 matrix difficult? | |
Ans. Finding the inverse of a 3x3 matrix involves several steps, including calculating the determinant, matrix of minors, matrix of cofactors, and adjugate matrix. While the process may seem complex at first, with practice and proper understanding of the steps involved, it becomes easier. Using calculators or software programs specifically designed for matrix operations can also simplify the process.
Text Transcript from Video
hello friends this video on matrices
part 37 is brought to you by exam viacom
no more fear from exam before watching
this video please make sure that you
have watched part 1 to part 36 let's
take one example where we have to again
find reverse using elementary
transformation here instead of two by
two is three by three matrix same thing
will apply here too
so let's assume this is a so we have the
formula a into a inverse physical I or
we can say that this line you know - one
five one three zero one three this into
a inverse is equal to I I will be this
time this guy 3 cross 3 matrix
why because everything is 3 cross 3
correct now if you see the first thing
you should do is we should convert this
to 1 because you finally make this as
I'm reading this guy should become 1 and
now we'll using via will be do using
your own transformation in this case I
like how it's simple for me
because everything is horizontal and we
make changes so to make this as 1 I can
simply divide the whole thing by 2 so
this becomes 1 so let's do this so if
you fast the division here because
Laemmle pushman if you take more space
okay so I'll say R 1 is nothing but R
unreactive this is my Rho 1 go to let me
write here also no 1 right this guy has
a RG divided by 2 this becomes 2 by 2
that is 1 0 by 2 0 minus 1 by 2
compared to other elements are still
Seng 0 which is 1 this is this is my
element on this side in a inverse is
equal to same thing happened over here
also this room I have to modify this
store should become 1 by 2 of this so 1
becomes 1 by 2 zero income 0 by 2 is you
become 0 by 2 and other rows are still
same no change now the second thing I do
is I have to activate alright in case of
rows we generally follow the clockwise
direction so next is just great this
guy's my target first time this way on a
target non this method I have to make it
0 to make it 0 I can't use the 0 I can
multiply with 0 that is not possible
because the operations of combated what
I get into is I can subtract this guy
with just go out this way so we subtract
5 with this guys doesn't make sense
because all 0 if you multiply this
number and subtract it's all 0 it won't
make much difference but you multiply
this guy number 5 and subtract 5 minus 5
will become 0 correct so I can say that
R 2 is nothing but R 2 minus 5 Arthur
why why because 5 minus 5 is 0 so let me
do that other elements are correct the
only element that makes it influences
are 2 so 5 minus 5 is 0 1 minus 5 times
0 is again 0 so 1 minus 0 is 1 and 0
minus minus 5 by 2 so right here is like
this 0 minus 5 into minus maybe this
whole thing's become 5 by 2 this becomes
a vital
because - - become plus and this row is
again changing this into a inverse is
nothing bad
hey that's what will change Row two so
one is unchanged Row two is r2 minus 5r1
so 0 minus 5 into 1 by 2 becomes 4 5
minus 5 by 2 1 minus 5m 0 does not make
sense
all will change this is mine so I have
these values set now my next target is
this guy this guy is 1 now this guy is 1
and this guy should be 0 how do I get 0
again even if I subtract with this it
won't make difference because 0 here I
have to subtract the road to only
subtract with Row 2 it will be 0 so I
can say that R 3 is nothing but R 3
minus R T why this guy might have this
guy becomes so other things working in
pattern but this is - this is 0 this is
already set so I can say like this other
values are not impacted R 3 is impacted
0-0 subtracting this from this 0 1 minus
1 0 3 minus 5 by 2 is 1 by 2 correct 3
months 5 between biting in a in words
this place where - again the first and
second row is not impacted thus this guy
is impacted 0 minus minus Phi by 2 is
plus PI by 2 and this guy is 0 minus 1
this is 1 0 minus 1 is minus 1 and this
guy is 1 minus 0 is again 1 oh this is
why I'm interested
these numbers are done now my next
target is this because these are
immature set number me this has to be
one this is very simple again just
multiply this by two so this this
knowing fact if you multiply if you
multiply our 3 by 2 this becomes 1 this
has going back so I can see there are 3
is nothing but twice Hartley so what I
get here is of an art you know impact
our 3 will have impact 0 into 2 is 0 0 2
2 is 0 0 into 1 by 2 is 1 a inverse is
nothing but here also Argonaut ammonium
van R 3 will have impact plus PI by 2
into 2 is 5 - 20 - mine - 2 is minus 2
and mine to do is this is what I am
going now I have to make this guy is 0
because this has to be 0 and this guy
also has to be so let me first make this
guy answer how to make this guy is what
I can do is and also I do not want to
impact these guys should I use these are
1 these basically impactive because some
additional come here if I use r3 these
guys won't have impact start because r3
into some number plus r2 this guys won't
happen because this gains are 0 so what
I can say that I can say are two there's
nothing but R 2 minus PI by 81 so if I
get you are cheap 5 by 2 minus 5 by 2
because this is mine to PI minus PI by 2
Phi by 2 minus PI by 2 VDC so what I get
here is 0 you mine so 1 0 minus 1 by 2
and art is getting impacted so this
becomes this guy becomes 0 minus 5 by
you will do 0 0 ly user
you can sm and 1 minus Phi by 2 0 equals
0 so this is 0 1 minus 0 is 0 this guy
against PI by 2 minus 1 into 5 by 2 I
can show you a way to minus 1 something
about 5 by 2 into PI by 2 so this
becomes this is 1 actually going finally
1 minus and this line is also on having
any back this into a inverse is no he
but they also first oh no impact third
row no impact the only backless second
one second row is nothing but 5 by 2
minus RT so this is 0 minus 5 by 2 in 2
or 3 so if you see this guy currently is
minus 5 by 3 this is nothing but minus
PI by 2 minus PI by 2 into r3
adres 5 this is nothing but minus 5 by 2
minus 25 way that this minus 30 by 2
that is minus 1 correct how I got see
this current value is minus PI by 2 this
has to be this minus PI by 2 into this
that is minus 15 space - this guy is 0
minus PI by 2 into this 0 minus PI by 2
n minus 3 so this becomes 5 this
reversible error so this becomes 0 minus
PI by 2 into this is 2 here this becomes
n so this guy a business so everything
here except this guy this guy here is
what minus 1 by 2 I am to make it what I
have to make it 0 what I can do is I can
I cannot use this one because this is 0
not have to do I can use object
what I can say is r1 is nothing we're on
one class this is one by two
if you make it one by T and you add this
this is from 0 so R 1 plus 1 by 2 into R
so then this becomes 0 because minus 1
by 2 plus 1 material comes in it so
let's do that so this is 0 so this won't
have any impact this won't have any time
this becomes minus 1 by 2 plus 1 by this
name is 1/2
does it have any impact this inverse is
you do here the first will have in fact
however is going to R 1 plus 1 by 2 R 3
so this becomes 1 by 2 plus 1 by 2 into
5 by 3 that this 1 by 2 plus 1 by 2 5
there is 5 by 2 plus 1 by 2 that is 6 by
2 the disc 3 similarly this guy will
become 0 plus 1 by 2 into minus 2 so
this is 1 by 2 into minus 2 is 1 plus 0
minus 1 so this is our instance saving
this guy also 0 plus 1 by 2 in game
write 0 plus 1 by 2 into 2 this is
cancel this is one see these rows are
not impacted so this is my past why
because this I bought I so I can say I a
inverse X this value so what I have done
here here is I've used row operations
but the logic here if you see I started
with this element I took these elements
I to this element I hope this event and
then I took these elements like this so
always follow general the convention is
anti-clockwise direction if you're using
row operation for column follow the
progress direction thank you visit exam
viacom to watch free education videos
try free online tests get the best
quality study materials study from the
best tutors and mentors and maximum
thank sensory
part 37 is brought to you by exam viacom
no more fear from exam before watching
this video please make sure that you
have watched part 1 to part 36 let's
take one example where we have to again
find reverse using elementary
transformation here instead of two by
two is three by three matrix same thing
will apply here too
so let's assume this is a so we have the
formula a into a inverse physical I or
we can say that this line you know - one
five one three zero one three this into
a inverse is equal to I I will be this
time this guy 3 cross 3 matrix
why because everything is 3 cross 3
correct now if you see the first thing
you should do is we should convert this
to 1 because you finally make this as
I'm reading this guy should become 1 and
now we'll using via will be do using
your own transformation in this case I
like how it's simple for me
because everything is horizontal and we
make changes so to make this as 1 I can
simply divide the whole thing by 2 so
this becomes 1 so let's do this so if
you fast the division here because
Laemmle pushman if you take more space
okay so I'll say R 1 is nothing but R
unreactive this is my Rho 1 go to let me
write here also no 1 right this guy has
a RG divided by 2 this becomes 2 by 2
that is 1 0 by 2 0 minus 1 by 2
compared to other elements are still
Seng 0 which is 1 this is this is my
element on this side in a inverse is
equal to same thing happened over here
also this room I have to modify this
store should become 1 by 2 of this so 1
becomes 1 by 2 zero income 0 by 2 is you
become 0 by 2 and other rows are still
same no change now the second thing I do
is I have to activate alright in case of
rows we generally follow the clockwise
direction so next is just great this
guy's my target first time this way on a
target non this method I have to make it
0 to make it 0 I can't use the 0 I can
multiply with 0 that is not possible
because the operations of combated what
I get into is I can subtract this guy
with just go out this way so we subtract
5 with this guys doesn't make sense
because all 0 if you multiply this
number and subtract it's all 0 it won't
make much difference but you multiply
this guy number 5 and subtract 5 minus 5
will become 0 correct so I can say that
R 2 is nothing but R 2 minus 5 Arthur
why why because 5 minus 5 is 0 so let me
do that other elements are correct the
only element that makes it influences
are 2 so 5 minus 5 is 0 1 minus 5 times
0 is again 0 so 1 minus 0 is 1 and 0
minus minus 5 by 2 so right here is like
this 0 minus 5 into minus maybe this
whole thing's become 5 by 2 this becomes
a vital
because - - become plus and this row is
again changing this into a inverse is
nothing bad
hey that's what will change Row two so
one is unchanged Row two is r2 minus 5r1
so 0 minus 5 into 1 by 2 becomes 4 5
minus 5 by 2 1 minus 5m 0 does not make
sense
all will change this is mine so I have
these values set now my next target is
this guy this guy is 1 now this guy is 1
and this guy should be 0 how do I get 0
again even if I subtract with this it
won't make difference because 0 here I
have to subtract the road to only
subtract with Row 2 it will be 0 so I
can say that R 3 is nothing but R 3
minus R T why this guy might have this
guy becomes so other things working in
pattern but this is - this is 0 this is
already set so I can say like this other
values are not impacted R 3 is impacted
0-0 subtracting this from this 0 1 minus
1 0 3 minus 5 by 2 is 1 by 2 correct 3
months 5 between biting in a in words
this place where - again the first and
second row is not impacted thus this guy
is impacted 0 minus minus Phi by 2 is
plus PI by 2 and this guy is 0 minus 1
this is 1 0 minus 1 is minus 1 and this
guy is 1 minus 0 is again 1 oh this is
why I'm interested
these numbers are done now my next
target is this because these are
immature set number me this has to be
one this is very simple again just
multiply this by two so this this
knowing fact if you multiply if you
multiply our 3 by 2 this becomes 1 this
has going back so I can see there are 3
is nothing but twice Hartley so what I
get here is of an art you know impact
our 3 will have impact 0 into 2 is 0 0 2
2 is 0 0 into 1 by 2 is 1 a inverse is
nothing but here also Argonaut ammonium
van R 3 will have impact plus PI by 2
into 2 is 5 - 20 - mine - 2 is minus 2
and mine to do is this is what I am
going now I have to make this guy is 0
because this has to be 0 and this guy
also has to be so let me first make this
guy answer how to make this guy is what
I can do is and also I do not want to
impact these guys should I use these are
1 these basically impactive because some
additional come here if I use r3 these
guys won't have impact start because r3
into some number plus r2 this guys won't
happen because this gains are 0 so what
I can say that I can say are two there's
nothing but R 2 minus PI by 81 so if I
get you are cheap 5 by 2 minus 5 by 2
because this is mine to PI minus PI by 2
Phi by 2 minus PI by 2 VDC so what I get
here is 0 you mine so 1 0 minus 1 by 2
and art is getting impacted so this
becomes this guy becomes 0 minus 5 by
you will do 0 0 ly user
you can sm and 1 minus Phi by 2 0 equals
0 so this is 0 1 minus 0 is 0 this guy
against PI by 2 minus 1 into 5 by 2 I
can show you a way to minus 1 something
about 5 by 2 into PI by 2 so this
becomes this is 1 actually going finally
1 minus and this line is also on having
any back this into a inverse is no he
but they also first oh no impact third
row no impact the only backless second
one second row is nothing but 5 by 2
minus RT so this is 0 minus 5 by 2 in 2
or 3 so if you see this guy currently is
minus 5 by 3 this is nothing but minus
PI by 2 minus PI by 2 into r3
adres 5 this is nothing but minus 5 by 2
minus 25 way that this minus 30 by 2
that is minus 1 correct how I got see
this current value is minus PI by 2 this
has to be this minus PI by 2 into this
that is minus 15 space - this guy is 0
minus PI by 2 into this 0 minus PI by 2
n minus 3 so this becomes 5 this
reversible error so this becomes 0 minus
PI by 2 into this is 2 here this becomes
n so this guy a business so everything
here except this guy this guy here is
what minus 1 by 2 I am to make it what I
have to make it 0 what I can do is I can
I cannot use this one because this is 0
not have to do I can use object
what I can say is r1 is nothing we're on
one class this is one by two
if you make it one by T and you add this
this is from 0 so R 1 plus 1 by 2 into R
so then this becomes 0 because minus 1
by 2 plus 1 material comes in it so
let's do that so this is 0 so this won't
have any impact this won't have any time
this becomes minus 1 by 2 plus 1 by this
name is 1/2
does it have any impact this inverse is
you do here the first will have in fact
however is going to R 1 plus 1 by 2 R 3
so this becomes 1 by 2 plus 1 by 2 into
5 by 3 that this 1 by 2 plus 1 by 2 5
there is 5 by 2 plus 1 by 2 that is 6 by
2 the disc 3 similarly this guy will
become 0 plus 1 by 2 into minus 2 so
this is 1 by 2 into minus 2 is 1 plus 0
minus 1 so this is our instance saving
this guy also 0 plus 1 by 2 in game
write 0 plus 1 by 2 into 2 this is
cancel this is one see these rows are
not impacted so this is my past why
because this I bought I so I can say I a
inverse X this value so what I have done
here here is I've used row operations
but the logic here if you see I started
with this element I took these elements
I to this element I hope this event and
then I took these elements like this so
always follow general the convention is
anti-clockwise direction if you're using
row operation for column follow the
progress direction thank you visit exam
viacom to watch free education videos
try free online tests get the best
quality study materials study from the
best tutors and mentors and maximum
thank sensory
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Nov 21, 2024 Last updated |
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