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
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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:
- If two rows (or columns) of a determinant are interchanged, the value of the determinant changes its sign.
- If any two rows (or columns) of a determinant are identical, then the value of the determinant is zero.
- If all the elements of any row (or column) of a determinant are multiplied by a scalar k, then the value of the determinant is also multiplied by k.
- If each element of a row (or column) of a determinant is expressed as the sum of two terms, then the determinant can be expressed as the sum of two determinants.
| 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 if we have a system of linear equations in the form Ax = b, where A is the coefficient matrix, x is the variable vector, and b is the constant vector, then the solution can be found by taking the determinants of various matrices.
To solve for the variables, we calculate the determinants of the matrices obtained by replacing each column of the coefficient matrix with the constant vector b. Then, the solution for each variable is given by the ratio of the determinant of the corresponding matrix to the determinant of the coefficient matrix.
| 3. Can determinants be used to find the area of a triangle? | |
Ans. Yes, determinants can be used to find the area of a triangle. Given the coordinates of the three vertices of a triangle, we can form a matrix using these coordinates. By taking half the absolute value of the determinant of this matrix, we can find the area of the triangle.
The formula to find the area of a triangle using determinants is:
Area = 0.5 * |(x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2))|
Here, (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices of the triangle.
| 4. Can determinants be negative? | |
Ans. Yes, determinants can be negative. The sign of the determinant depends on the arrangement of the elements in the matrix. If the number of row or column interchanges required to transform the matrix into its row-echelon form is odd, then the determinant will be negative. If the number of row or column interchanges is even, the determinant will be positive.
For example, if we have a 2x2 matrix [a b; c d], the determinant is given by ad - bc. If ad - bc is negative, then the determinant itself is negative.
| 5. Are determinants only used in linear algebra? | |
Ans. No, determinants are not only used in linear algebra. While determinants have a significant role in linear algebra, they also find applications in various other fields, such as calculus, physics, statistics, and computer science.
In calculus, determinants are used to find the Jacobian matrix, which is used in multivariable calculus and change of variables.
In physics, determinants are used to calculate the moment of inertia of objects and to solve problems involving rotational motion.
In statistics, determinants are used in multivariate analysis, such as in the calculation of covariance matrices.
In computer science, determinants are used in graphics and image processing algorithms, such as transformations and image compression.
Text Transcript from Video
hello friends this video on determinants
part 15 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 14 and
solved without expand until tricky part
so I am here to prove that to be 0 the
rule is to be 0 we have to do that two
rows or two columns are identical but we
don't see any rows or columns identical
but the trick here is we can see that
these guys are big guys big in numbers
so if we can somehow break this in such
a fashion that you get 2 & 7 that is I
can write 65 as some key in to 7 plus 2
kind of all correct in that case what I
can do is I can break this determine
into multiple determinants and then I
can say that the columns are similar see
we can write 65 into K into some 7 form
and 75 also into some key into 7 for the
K into 8 K into 8 plus 3 and 86 also I
can write as some K into 9 plus 5 form
if you can write in this fashion then my
problem to solve for this type of thing
see or other way round also or other way
round in the sense this fashion he into
2 plus 7 or K into 3 plus 8 or K to 9
plus 5 in K into Phi plus name any of
this fashion is Paquito or in this
fashion awesome or in this fashion also
if I can write L to you 65 is equal to
XI ever need to do plus K 2 into 7 675
this is the I mean the best
representation actually 75 s K 1 into
this guy is 3 now plus
into eight and eighty-six also if I can
write as K 1 into 5 by 5 because this
guy is 5 now plus K 2 into 9 so they can
write in this fashion what will happen
is I can break this in two matrices and
then I an say that the first met with
also is 0 why because is 2 3 5 is here
and push me baby's here I will say it's
proportional to columns of proportions
of 0 the second determine we shall get
will also be 0 why because you see this
guy will be not like this 7 8 9 will be
and this guy will be proportional so I
can see that they are 0 correct see the
logic is if you can understand if I
break this end up in a what I will be
getting is 2 3 5 7 8 9 instead of 65 a
right this guy and I break this I will
get this as he were into two even in 3
and this way I will get K 1 into 5 the
whole determinants or this is straight
only this is straight my mistake he is
not met with his straight determinant
plus again same thing 2 3 5 7 8 9 he
into 7 he 2 into 8 and this guy is
creative in tonight
see what I am trying to say is I'm
trying to give you the approach to
follow so if I can break into this form
and I can say that this guy in this guy
proportional and here this guy is 7 8 9
and Katie 7 and proportional so they are
0 correct and thus it is visible such
question you have to break these numbers
so the logic here is we have to find the
value of K 1 and K 2 so let's try let's
have to equation because we have to
equation and see what is the value of K
1 K 2 you are getting and if that value
satisfies the third equation so there is
2 K 1 plus 7 K 2 is equal to 65 first
equation second equation see is that 3 K
1 plus 8
q is equal to 75 so let's solve this
equation to solve this equation what I
can do is and you multiply this guy by
to multiply this guy by 3 so what I get
here is this guy is 6 K 1 plus 21 K 2 is
equal to 195
this guy is 6 K 1 plus 16 kid is equal
to 150 subtract this what I get is 21
minus 16 is 5 K T is equal to 195 minus
50 is 45 or I get K 2 is equal to 9
once I got K 2 will let me find K 1 so I
can find any of the equations I got K 2
so I got 65 is equal to K 1 plus 7 into
K 2 K 2 is 9 correct on I get K 2 as
nothing bad 65 minus 9 into 7 63 by 2
that is 2 by 2 so I got K 2 as 9 and K 1
L 1 this is not the solution you have to
first shake with that that particular
key when K 2 satisfies this condition
also if this satisfies then only we can
break this into K 1 K 2 form so let's do
that
let's see here L is you have to find the
value of 5 K 1 plus 9 K 2 correct 592
and make sure it is 86 so let us find
the value 5 given 5 into 1 plus 9 2 K 2
K 2 is how much K 2 is we got 9 so that
is 9 81 + 5 86 and we expected 86
correct thus we got the value of K 1 K
so now what we can do is we can break
this guy into this form 7e 965 is
nothing but 2 ki 1
it is 2 into 1 plus plus 7 into K 2 that
is 7 into K 2 is how much 9 this guy 75
is 3 K 1 3 into 1 plus 8 K 289 this guy
I can write as 86 s 5k 1 5 into 1 plus K
2 into 9 that is K 2 is how much 9 so 9
into 9 so this now I can break as this
form 3 5 7 8 9 this guy comes to me 5
because I am breaking this guy I am
taking these guys now first Plus now I
dig these legs this becomes 2 3 5 7 8 9
and this becomes 7 into 9 and this guy
becomes 80 to 9 and this guy becomes 9
now if you see these two columns are
same so this is zip here also you see
these column and these column are sorry
not this this column and this column are
proportional because they are seven
seven two nine eight eight nine nine
nine tonight
so this guy's also so this is so I got
select so now here also if you see what
we have done we have solved this
equation without even expanding this
just by using power of properties so in
such questions the cut in such questions
the catch is that you have to find this
values K 1 K 2 says that you can break
these bigger numbers and then then break
this bigger determiner to smaller to the
minute and then you can use property
even that they are self
let's take one more example here also we
have to prove that it is zip so what
does the approach we can follow what can
we do
you don't know we are helpless the first
thing we can expand this so let's expand
this if we can add some clean so a TB
plus is av plus AC one CA this becomes V
CE plus V 1 AV C plus C me he just don't
focus on solving the question understand
that push why we are solving in that
fashion now we have to make it 0 to make
it zero one thing we know that we had to
show that two columns are two rows are
proportion you see this Row is all one
this roy's are different this was also
all different how can I make it
provision now if you observe you add
these two rows columns columns 2 and
column 3 a B plus BC plus C this becomes
a B plus BC plus C this also becomes a
be bloody secrecy all will become same
right here also all are same then we can
see that 2 columns are proportional and
then that means it is 0 so now the next
step is we have to apply C 3 is equal to
C 2 C 3 plus C 2 if you add these two
columns let's up because we want to do
so through that two columns are
proportional we saw this is 1 1 1 if you
can prove that all the numbers in a
particular columns are also equal that
means they're proportional that's 0 so
let's do that let's add these two
columns make it proportional so this
becomes 1 1 1 no change C 2 is also no
change bc c AV this guy is change this
becomes C 3 is equal to C 3 plus C T
that means you add these two guys so
this becomes a B plus BC plus C this
also becomes a B plus BC plus C another
becomes a B plus BC
see if we plums we seamlessly
correct now if you can see that these
two columns are proportional because or
again take a B plus BC plus C common
make life simple to show you in a better
way this becomes 1 1 1 because BC C a
and a B and this becomes 1 1 1 now if
you see these two columns are identical
since they are identical this becomes 0
cell this becomes 0 this becomes a B
plus BC plus C into into 0 and that is
big 0 that is my answer so what we have
done we just try to manually the column
to make sure that two columns are
identical and that's why I have done all
identical are proportional so we just
added column 3 is equal to come 3 plus
column 2 and I found that column 3 and
column 1 we came for washed up and then
the whole Peter a million became say
will take a genome 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 serene
part 15 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 14 and
solved without expand until tricky part
so I am here to prove that to be 0 the
rule is to be 0 we have to do that two
rows or two columns are identical but we
don't see any rows or columns identical
but the trick here is we can see that
these guys are big guys big in numbers
so if we can somehow break this in such
a fashion that you get 2 & 7 that is I
can write 65 as some key in to 7 plus 2
kind of all correct in that case what I
can do is I can break this determine
into multiple determinants and then I
can say that the columns are similar see
we can write 65 into K into some 7 form
and 75 also into some key into 7 for the
K into 8 K into 8 plus 3 and 86 also I
can write as some K into 9 plus 5 form
if you can write in this fashion then my
problem to solve for this type of thing
see or other way round also or other way
round in the sense this fashion he into
2 plus 7 or K into 3 plus 8 or K to 9
plus 5 in K into Phi plus name any of
this fashion is Paquito or in this
fashion awesome or in this fashion also
if I can write L to you 65 is equal to
XI ever need to do plus K 2 into 7 675
this is the I mean the best
representation actually 75 s K 1 into
this guy is 3 now plus
into eight and eighty-six also if I can
write as K 1 into 5 by 5 because this
guy is 5 now plus K 2 into 9 so they can
write in this fashion what will happen
is I can break this in two matrices and
then I an say that the first met with
also is 0 why because is 2 3 5 is here
and push me baby's here I will say it's
proportional to columns of proportions
of 0 the second determine we shall get
will also be 0 why because you see this
guy will be not like this 7 8 9 will be
and this guy will be proportional so I
can see that they are 0 correct see the
logic is if you can understand if I
break this end up in a what I will be
getting is 2 3 5 7 8 9 instead of 65 a
right this guy and I break this I will
get this as he were into two even in 3
and this way I will get K 1 into 5 the
whole determinants or this is straight
only this is straight my mistake he is
not met with his straight determinant
plus again same thing 2 3 5 7 8 9 he
into 7 he 2 into 8 and this guy is
creative in tonight
see what I am trying to say is I'm
trying to give you the approach to
follow so if I can break into this form
and I can say that this guy in this guy
proportional and here this guy is 7 8 9
and Katie 7 and proportional so they are
0 correct and thus it is visible such
question you have to break these numbers
so the logic here is we have to find the
value of K 1 and K 2 so let's try let's
have to equation because we have to
equation and see what is the value of K
1 K 2 you are getting and if that value
satisfies the third equation so there is
2 K 1 plus 7 K 2 is equal to 65 first
equation second equation see is that 3 K
1 plus 8
q is equal to 75 so let's solve this
equation to solve this equation what I
can do is and you multiply this guy by
to multiply this guy by 3 so what I get
here is this guy is 6 K 1 plus 21 K 2 is
equal to 195
this guy is 6 K 1 plus 16 kid is equal
to 150 subtract this what I get is 21
minus 16 is 5 K T is equal to 195 minus
50 is 45 or I get K 2 is equal to 9
once I got K 2 will let me find K 1 so I
can find any of the equations I got K 2
so I got 65 is equal to K 1 plus 7 into
K 2 K 2 is 9 correct on I get K 2 as
nothing bad 65 minus 9 into 7 63 by 2
that is 2 by 2 so I got K 2 as 9 and K 1
L 1 this is not the solution you have to
first shake with that that particular
key when K 2 satisfies this condition
also if this satisfies then only we can
break this into K 1 K 2 form so let's do
that
let's see here L is you have to find the
value of 5 K 1 plus 9 K 2 correct 592
and make sure it is 86 so let us find
the value 5 given 5 into 1 plus 9 2 K 2
K 2 is how much K 2 is we got 9 so that
is 9 81 + 5 86 and we expected 86
correct thus we got the value of K 1 K
so now what we can do is we can break
this guy into this form 7e 965 is
nothing but 2 ki 1
it is 2 into 1 plus plus 7 into K 2 that
is 7 into K 2 is how much 9 this guy 75
is 3 K 1 3 into 1 plus 8 K 289 this guy
I can write as 86 s 5k 1 5 into 1 plus K
2 into 9 that is K 2 is how much 9 so 9
into 9 so this now I can break as this
form 3 5 7 8 9 this guy comes to me 5
because I am breaking this guy I am
taking these guys now first Plus now I
dig these legs this becomes 2 3 5 7 8 9
and this becomes 7 into 9 and this guy
becomes 80 to 9 and this guy becomes 9
now if you see these two columns are
same so this is zip here also you see
these column and these column are sorry
not this this column and this column are
proportional because they are seven
seven two nine eight eight nine nine
nine tonight
so this guy's also so this is so I got
select so now here also if you see what
we have done we have solved this
equation without even expanding this
just by using power of properties so in
such questions the cut in such questions
the catch is that you have to find this
values K 1 K 2 says that you can break
these bigger numbers and then then break
this bigger determiner to smaller to the
minute and then you can use property
even that they are self
let's take one more example here also we
have to prove that it is zip so what
does the approach we can follow what can
we do
you don't know we are helpless the first
thing we can expand this so let's expand
this if we can add some clean so a TB
plus is av plus AC one CA this becomes V
CE plus V 1 AV C plus C me he just don't
focus on solving the question understand
that push why we are solving in that
fashion now we have to make it 0 to make
it zero one thing we know that we had to
show that two columns are two rows are
proportion you see this Row is all one
this roy's are different this was also
all different how can I make it
provision now if you observe you add
these two rows columns columns 2 and
column 3 a B plus BC plus C this becomes
a B plus BC plus C this also becomes a
be bloody secrecy all will become same
right here also all are same then we can
see that 2 columns are proportional and
then that means it is 0 so now the next
step is we have to apply C 3 is equal to
C 2 C 3 plus C 2 if you add these two
columns let's up because we want to do
so through that two columns are
proportional we saw this is 1 1 1 if you
can prove that all the numbers in a
particular columns are also equal that
means they're proportional that's 0 so
let's do that let's add these two
columns make it proportional so this
becomes 1 1 1 no change C 2 is also no
change bc c AV this guy is change this
becomes C 3 is equal to C 3 plus C T
that means you add these two guys so
this becomes a B plus BC plus C this
also becomes a B plus BC plus C another
becomes a B plus BC
see if we plums we seamlessly
correct now if you can see that these
two columns are proportional because or
again take a B plus BC plus C common
make life simple to show you in a better
way this becomes 1 1 1 because BC C a
and a B and this becomes 1 1 1 now if
you see these two columns are identical
since they are identical this becomes 0
cell this becomes 0 this becomes a B
plus BC plus C into into 0 and that is
big 0 that is my answer so what we have
done we just try to manually the column
to make sure that two columns are
identical and that's why I have done all
identical are proportional so we just
added column 3 is equal to come 3 plus
column 2 and I found that column 3 and
column 1 we came for washed up and then
the whole Peter a million became say
will take a genome 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 serene
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About this Video
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Dec 22, 2024 Last updated |
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