JEE Exam > JEE Videos > Mathematics (Maths) Class 12 > Examples : Based on Properties of Determinants
Examples : Based on Properties of Determinants Video Lecture | Mathematics (Maths) Class 12 - JEE
FAQs on Examples : Based on Properties of Determinants Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What are the properties of determinants? |  |
Ans. The properties of determinants include:
- Multiplying a row or column by a scalar multiplies the determinant by the same scalar.
- Swapping two rows or columns changes the sign of the determinant.
- Adding a multiple of one row or column to another leaves the determinant unchanged.
- If a matrix has a row or column of zeros, then its determinant is zero.
- If two rows or columns of a matrix are equal, then its determinant is zero.
| 2. How can determinants be used to solve systems of linear equations? | |
Ans. Determinants can be used to solve systems of linear equations by using Cramer's Rule. Cramer's Rule states that the solution to a system of linear equations can be obtained by dividing the determinants of matrices formed from the coefficients of the variables. By evaluating the determinants and dividing them accordingly, the values of the variables can be found.
| 3. Can determinants be negative numbers? | |
Ans. Yes, determinants can be negative numbers. The sign of the determinant depends on the properties of the matrix. Swapping two rows or columns changes the sign of the determinant, so if a matrix has an odd number of row or column swaps, the determinant will be negative. If it has an even number of swaps, the determinant will be positive.
| 4. What happens if the determinant of a matrix is zero? | |
Ans. If the determinant of a matrix is zero, it means that the matrix is singular or non-invertible. This means that the matrix does not have a unique solution and the system of linear equations it represents may have either infinitely many solutions or no solution at all. This is an important condition to check when solving systems of equations using determinants.
| 5. Can determinants be used to find areas or volumes? | |
Ans. Yes, determinants can be used to find areas or volumes. In two-dimensional space, the absolute value of the determinant of a 2x2 matrix represents the area of the parallelogram formed by the column vectors of the matrix. In three-dimensional space, the absolute value of the determinant of a 3x3 matrix represents the volume of the parallelepiped formed by the column vectors of the matrix. By using determinants, geometric properties such as areas and volumes can be calculated.
Text Transcript from Video
hello friends this video on determinants
part 16 is brought you by exam fyodor
harm no more fear from exam before
watching this video please make sure
that you have watched part one two parts
of teen sample this please pay attention
because all these questions where we
have to solve without expanding they
have to think a lot see from this
example what we can make out is in the
right hand side we have elements such as
a BP CRX all single elements the left
hand side we have everything on addition
so that means we have to somehow break
this without breaking records on right
if you break this then only you'll get
signal limit for example if you break
this guy this becomes B you come see so
you will get single elements let's break
this and then solve this should break
the left-hand sides I'm taking LHS only
now so what you get this let me break
this guy fast break these values this
becomes Q plus R this is all plus P this
will become P I plus Z is n plus X X
just hidden trial method because we know
we have to break so let's break this
that's all we are doing and so also we b
plus c c plus a you can try breaking in
some other form we should be able to get
answered but the logic was here to break
it unless we broke it becomes y plus X
and this is X this is y this is my
determinate I got this determinant as
sum of two trim minutes correct now what
should we do now we have to make if you
see AP X is here we have this BP y is
here that means somehow we have got one
row in both of these I mean just think
this our thought process because we got
similar
heartily same no which we looking for
same row which we are looking for here
right see up and down weekend interest
and begin swap rows that also we have
property but we have got exact same row
and that is if we can somehow convert
these guys also in exact same rows
column is solve because if you see a px
this guy is similar to this guy and vqi
similar to this way so that is even
convert this guy into these these rows
Row 1 and Row 2 it for this one into
this fashion a column salt now if you
see if you complete these two rows if
you say C plus a minus a you get c r+ p
- r u8 r z+ x - x you get Z so you add c
r NZ and c are in the d is something
which were looking for correct that is
if you apply if you if you say in this
case in this case if you say C 2 is
equal to C 2 minus C 3 sorry R 2 is
equal to R 2 minus R 3 Row 2 is going to
do Row 2 minus Row 3 so what you get you
get again a similar row for this first
one and for second one if you say let us
do the same thing for you we have got B
here I can supply B from this but I can
subtract B from these row first row so
here I say R 1 is nothing but R 1 minus
R 3 so in this al gets see this I will
get R and this I will get Z see our
asset the one I am looking for so what I
will get here V plus C minus on B plus C
is not change here because this for this
it is R 2 is equal to R 2 minus R 3 so R
1 is not changed so let's find our 1
here like this V plus C Q plus R Y plus
there are 1 become R 2 becomes R 2 minus
R 3 C plus a minus a becomes C R plus P
minus P R Z plus X minus that Z and R 3
is same 80s
class for this I am applying r1 is equal
to gamma minus r3 so B plus C minus VC Q
plus R minus Q is our y plus Z minus y Z
my r2 is unchanged
and mine r3 is also unchanged this is
what we have got now what do you see so
we have got two rows here and two rows
here can we convert this guy also in one
of the rows yes we can you see this is B
plus C if I subtract this guy from this
guy B plus C minus C become B Q plus R
minus R become Q and y plus Z minus Z
become Y so this becomes B Q Y this the
one I am looking for similarly here also
I have C plus ay R + P + y plus X if I
subtract C I can subtract this from this
because there is no common but C plus a
minus C I get a R plus P minus R I get P
Y plus X this is y plus X minus y I am
NOT able to subtract this this is C R
and this is z r+ p in this y plus Z plus
X actually correct Z plus X Z plus X
minus Z I get fix CRX is something one
more pair I am looking for so let's do
this
so here what I do I'll perform the
operation so this guy is this one so
here I am going to perform the operation
is R 1 is nothing but R 1 minus R 2 and
here I am going to say R 2 is nothing
but R two minus R one correct R 2 is R 2
minus R 1 here I 1 is R 1 minus R 2 so
what I get here is B plus C minus C is B
Q plus R minus RS q y plus Z minus Z is
y and other rows are unchanged C are
at AP X similarly glass first flow is
unchanged because no change in row 1
second row I am changing R 2 is equal to
R 2 minus R 1 C plus a minus C is a r+ p
- r SP z + X - that is X and the last is
also unchanged V Q now if you see I
bought two determinants and the values
are look sames bqi i have 1 CR XZ there
a px is there here also c RZ I have a px
I have BQ I am but it is not in the same
pattern
why because I want a px to main top bqi
in second and c RZ inside so I can
perform one operation where I can move
this guy to this guy right AP X n this
guy r2 + r3 again spam so ap has come
first and then I can swear after I am
done with that I can swap this guy with
this guy because if you see here what I
am trying to do is I am trying to get
T's okay let me try one operation first
so let me first a px in the top position
in this guy also let me try to put a px
in the first position because I want a
piece in the first position so with this
what I get is here the rule I am
following is I am swapping Row 1 and Row
3 here I'm swapping a row 1 end correct
so what I get is in this a px become
first and then cr-z is unchanged and
then vqi for this what I get is again a
px become first because I change this
this guy and cr-z came here and vqi is
in the bottom
but what I am looking for is a PX v qy
and cr-z so I have to swab this guy's
not face fab these guys I'm in good
condition
well so have these guys I am in good
condition correct so let me slap that so
that we I am swapping here are two with
our three in both kisses so what I get
is and they are same Ashley so I can
just say two into KP x vq y CR and that
is my answer so what I've done this step
I swapped also and I added ups because
they are same so as we do into this and
I swap R 2 and R 3 and I get incense and
this is my answer so this was a little
tricky question where you think a lot
while you are doing this the first clue
which we got was because here we have
all separated and here we have all added
so we broke it the moment we broke it we
found that we got this this this row and
this row the one which you're looking
for and then we found that if you
subtract this row from this row will
give a more row which we are looking for
we got two rows which we are looking for
and then we found that for the list also
we subtract with this guy really similar
kind of flow but in that we got all the
rows which we are looking for but they
were not arranged properly so we arrange
them properly is in property and we
heard the let's take one more example
here house of either have to solve
without expanding it see here also you
have to somehow prove that two rows or
columns are identical how to through
this I don't know we difficult question
actually if you see this next let's try
to prove one on these rows as bcz what
does one approach one approach I can
think of is if I can multiply these
numbers with C and this B so both this
guy will have BC and if I subtract these
two it will become 0 and then here also
I may get the similar value just trying
so let me multiply this guy with C and
this guy even this guy will see in this
guy with D why I am doing this because I
get minus BC I get minus BC here they'll
try to subtract correct so this is
nothing but 1 by BC into all multiply
the first row with C so this becomes 0
into C is 0 into C is AC becomes minus
PC minus a into B is minus AV 0 minus BC
BC NZ but I am done R 1 is nothing but C
R 1 and R 2 is nothing but we are now
I'll try to subtract this because that
gives me 0 there I'm trying to see if I
get equal so subtract these two value 8
1 by VC into 0 minus a B is EB AC minus
0 is a C and minus BC minus minus BC s Z
so what I have done here
I told R 1 is nothing but R 1 minus part
and everything else will be safe this
way will be BC correct
everything is insane now if you see they
are proportional 0 0 C CA B ba correct
so let me take out a from this so this
becomes a B by C this becomes B C 0 this
becomes minus kV 0 minus BC this becomes
please easy so what I have done I have
told R 1 is nothing but R 1 by a so I am
Telling this I took out a from here now
if you see this guy and this guy say so
this guy is nothing but a device into 0
and that is it so the target here was to
make to Rho 0 and somehow we are strided
we just saw and we thought ok let me
multiply the first guy with B and
settings I will see and subtract them to
get 0 and got the answer you can try
some other way also you get then so if
you just keep trying different
properties and
logically you get the answer with the
intention that you have to make to row
or to column
identical 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 much more
thanks and Satine
part 16 is brought you by exam fyodor
harm no more fear from exam before
watching this video please make sure
that you have watched part one two parts
of teen sample this please pay attention
because all these questions where we
have to solve without expanding they
have to think a lot see from this
example what we can make out is in the
right hand side we have elements such as
a BP CRX all single elements the left
hand side we have everything on addition
so that means we have to somehow break
this without breaking records on right
if you break this then only you'll get
signal limit for example if you break
this guy this becomes B you come see so
you will get single elements let's break
this and then solve this should break
the left-hand sides I'm taking LHS only
now so what you get this let me break
this guy fast break these values this
becomes Q plus R this is all plus P this
will become P I plus Z is n plus X X
just hidden trial method because we know
we have to break so let's break this
that's all we are doing and so also we b
plus c c plus a you can try breaking in
some other form we should be able to get
answered but the logic was here to break
it unless we broke it becomes y plus X
and this is X this is y this is my
determinate I got this determinant as
sum of two trim minutes correct now what
should we do now we have to make if you
see AP X is here we have this BP y is
here that means somehow we have got one
row in both of these I mean just think
this our thought process because we got
similar
heartily same no which we looking for
same row which we are looking for here
right see up and down weekend interest
and begin swap rows that also we have
property but we have got exact same row
and that is if we can somehow convert
these guys also in exact same rows
column is solve because if you see a px
this guy is similar to this guy and vqi
similar to this way so that is even
convert this guy into these these rows
Row 1 and Row 2 it for this one into
this fashion a column salt now if you
see if you complete these two rows if
you say C plus a minus a you get c r+ p
- r u8 r z+ x - x you get Z so you add c
r NZ and c are in the d is something
which were looking for correct that is
if you apply if you if you say in this
case in this case if you say C 2 is
equal to C 2 minus C 3 sorry R 2 is
equal to R 2 minus R 3 Row 2 is going to
do Row 2 minus Row 3 so what you get you
get again a similar row for this first
one and for second one if you say let us
do the same thing for you we have got B
here I can supply B from this but I can
subtract B from these row first row so
here I say R 1 is nothing but R 1 minus
R 3 so in this al gets see this I will
get R and this I will get Z see our
asset the one I am looking for so what I
will get here V plus C minus on B plus C
is not change here because this for this
it is R 2 is equal to R 2 minus R 3 so R
1 is not changed so let's find our 1
here like this V plus C Q plus R Y plus
there are 1 become R 2 becomes R 2 minus
R 3 C plus a minus a becomes C R plus P
minus P R Z plus X minus that Z and R 3
is same 80s
class for this I am applying r1 is equal
to gamma minus r3 so B plus C minus VC Q
plus R minus Q is our y plus Z minus y Z
my r2 is unchanged
and mine r3 is also unchanged this is
what we have got now what do you see so
we have got two rows here and two rows
here can we convert this guy also in one
of the rows yes we can you see this is B
plus C if I subtract this guy from this
guy B plus C minus C become B Q plus R
minus R become Q and y plus Z minus Z
become Y so this becomes B Q Y this the
one I am looking for similarly here also
I have C plus ay R + P + y plus X if I
subtract C I can subtract this from this
because there is no common but C plus a
minus C I get a R plus P minus R I get P
Y plus X this is y plus X minus y I am
NOT able to subtract this this is C R
and this is z r+ p in this y plus Z plus
X actually correct Z plus X Z plus X
minus Z I get fix CRX is something one
more pair I am looking for so let's do
this
so here what I do I'll perform the
operation so this guy is this one so
here I am going to perform the operation
is R 1 is nothing but R 1 minus R 2 and
here I am going to say R 2 is nothing
but R two minus R one correct R 2 is R 2
minus R 1 here I 1 is R 1 minus R 2 so
what I get here is B plus C minus C is B
Q plus R minus RS q y plus Z minus Z is
y and other rows are unchanged C are
at AP X similarly glass first flow is
unchanged because no change in row 1
second row I am changing R 2 is equal to
R 2 minus R 1 C plus a minus C is a r+ p
- r SP z + X - that is X and the last is
also unchanged V Q now if you see I
bought two determinants and the values
are look sames bqi i have 1 CR XZ there
a px is there here also c RZ I have a px
I have BQ I am but it is not in the same
pattern
why because I want a px to main top bqi
in second and c RZ inside so I can
perform one operation where I can move
this guy to this guy right AP X n this
guy r2 + r3 again spam so ap has come
first and then I can swear after I am
done with that I can swap this guy with
this guy because if you see here what I
am trying to do is I am trying to get
T's okay let me try one operation first
so let me first a px in the top position
in this guy also let me try to put a px
in the first position because I want a
piece in the first position so with this
what I get is here the rule I am
following is I am swapping Row 1 and Row
3 here I'm swapping a row 1 end correct
so what I get is in this a px become
first and then cr-z is unchanged and
then vqi for this what I get is again a
px become first because I change this
this guy and cr-z came here and vqi is
in the bottom
but what I am looking for is a PX v qy
and cr-z so I have to swab this guy's
not face fab these guys I'm in good
condition
well so have these guys I am in good
condition correct so let me slap that so
that we I am swapping here are two with
our three in both kisses so what I get
is and they are same Ashley so I can
just say two into KP x vq y CR and that
is my answer so what I've done this step
I swapped also and I added ups because
they are same so as we do into this and
I swap R 2 and R 3 and I get incense and
this is my answer so this was a little
tricky question where you think a lot
while you are doing this the first clue
which we got was because here we have
all separated and here we have all added
so we broke it the moment we broke it we
found that we got this this this row and
this row the one which you're looking
for and then we found that if you
subtract this row from this row will
give a more row which we are looking for
we got two rows which we are looking for
and then we found that for the list also
we subtract with this guy really similar
kind of flow but in that we got all the
rows which we are looking for but they
were not arranged properly so we arrange
them properly is in property and we
heard the let's take one more example
here house of either have to solve
without expanding it see here also you
have to somehow prove that two rows or
columns are identical how to through
this I don't know we difficult question
actually if you see this next let's try
to prove one on these rows as bcz what
does one approach one approach I can
think of is if I can multiply these
numbers with C and this B so both this
guy will have BC and if I subtract these
two it will become 0 and then here also
I may get the similar value just trying
so let me multiply this guy with C and
this guy even this guy will see in this
guy with D why I am doing this because I
get minus BC I get minus BC here they'll
try to subtract correct so this is
nothing but 1 by BC into all multiply
the first row with C so this becomes 0
into C is 0 into C is AC becomes minus
PC minus a into B is minus AV 0 minus BC
BC NZ but I am done R 1 is nothing but C
R 1 and R 2 is nothing but we are now
I'll try to subtract this because that
gives me 0 there I'm trying to see if I
get equal so subtract these two value 8
1 by VC into 0 minus a B is EB AC minus
0 is a C and minus BC minus minus BC s Z
so what I have done here
I told R 1 is nothing but R 1 minus part
and everything else will be safe this
way will be BC correct
everything is insane now if you see they
are proportional 0 0 C CA B ba correct
so let me take out a from this so this
becomes a B by C this becomes B C 0 this
becomes minus kV 0 minus BC this becomes
please easy so what I have done I have
told R 1 is nothing but R 1 by a so I am
Telling this I took out a from here now
if you see this guy and this guy say so
this guy is nothing but a device into 0
and that is it so the target here was to
make to Rho 0 and somehow we are strided
we just saw and we thought ok let me
multiply the first guy with B and
settings I will see and subtract them to
get 0 and got the answer you can try
some other way also you get then so if
you just keep trying different
properties and
logically you get the answer with the
intention that you have to make to row
or to column
identical 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 much more
thanks and Satine
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Oct 23, 2025
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