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Syllabus

Using properties of determinants prove that -

(b+c)

^{2}....a^{2}........a^{2}b

^{2}.....(c+a)^{2.}.....b^{2}=2abc(a+b+c)^{3}c

^{2}.....c^{2}.......(a+b)^{2}In this ques.. i just want to know tht after applying C

_{1}→ C_{1}-C_{2}, C_{2}→ C_{2}-C_{3}in this ques how can i take (a+b+c) common from C

_{1}and C_{2}.if A is a square matrix of order 3, such that / adj.A / = 64 . then find / A' / .

[1 0 1] [c-b c+a a-b]

[1 1 0] [b-c a-c a+b]

show that ABA

^{-1 }is a diagonal matrix .Prove that

| (b+c)^2 a^2 a^2 |

| b^2 (c+a)^2 b^2 | = 2abc(a+b+c)^3

| c^2 c^2 (a+b)^2 |

^{3}- b^{3}-c^{3}If a,b,c, all positive ,are pth,qth and rth terms of G.P. , prove that determinant [ log a p 1

log b q 1 = 0

log c r 1 ]

Difference between cramer's rule and Matrix method.....and when to use which one.....

if a is a square matrix of order 3 and / 3A / = k/A/ find value of k? pls fast plss

5.Three schools A, B and C want to award their selected students for the values of honesty, regularity and hard work. Each school decided to award a sum of Rs. 2500, Rs. 3100, Rs. 5100 per student for the respective values. The number of students to be awarded by the three schools as given below:A = 50500, 40800, 41600

Prove that the following determinant is equal to (ab + bc + ca)

^{3 :}-bc b

^{2}+ bc c^{2}+ bca

^{2}+ ac -ac c^{2}+ aca

^{2}+ ab b^{2}+ ab -ab|2 y 3|

|1 1 z|

xyz=80 and 3x+2y+10z=20

Find value of A(adjA)

A matrix of order 3X3 has determinant 5. What is the value of |3A|?

Q.(ii) If A = $\left|\begin{array}{ccc}5& 6& -3\\ -4& 3& 2\\ -4& -7& 3\end{array}\right|$, then write the cofactor of the element ${a}_{21}$ of its 2nd row.

^{T}|1. Using properties of determinants, prove the following:

| x y z

x

^{2}y^{2}z^{2}x

^{3}y^{3}z^{3 | = }xyz(x - y)(y - z)(z - x) .2. Using properties of determinants, prove the following :

| x x

^{2}1+px^{3}y y

^{2}1+py^{3}z z

^{2}1+pz^{3}| = (1+ pxyz)(x - y)(y - z)(z - x) .13.

$\mathrm{If}\mathrm{A}+\mathrm{B}+\mathrm{C}=\mathrm{\pi},\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left|\begin{array}{ccc}\mathrm{sin}\left(\mathrm{A}+\mathrm{B}+\mathrm{C}\right)& \mathrm{sin}\mathrm{B}& \mathrm{cost}\mathrm{C}\\ -\mathrm{sin}\mathrm{A}& 0& \mathrm{tan}\mathrm{A}\\ \mathrm{cos}\left(\mathrm{A}+\mathrm{B}\right)& -\mathrm{tan}\mathrm{A}& 0\end{array}\right|\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Using}\mathrm{propertles}\mathrm{of}\mathrm{determinant},\mathrm{prove}\mathrm{that}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left|\begin{array}{ccc}\mathrm{b}+\mathrm{c}& \mathrm{a}-\mathrm{b}& \mathrm{a}\\ \mathrm{c}+\mathrm{a}& \mathrm{b}-\mathrm{c}& \mathrm{b}\\ \mathrm{a}+\mathrm{b}& \mathrm{c}-\mathrm{a}& \mathrm{c}\end{array}\right|=3\mathrm{abc}-{\mathrm{a}}^{3}-{\mathrm{b}}^{3}-{\mathrm{c}}^{3}$

If det [ p b c

a q c = 0 then find (p/p-a) + (q/q-b) + (r/r-c)

a b r]

Given I

_{2}. Find determinant I_{2}. also find determinant 3I_{2}.PROVE THAT THE DETERMINANT

b

^{2}+c^{2}ab acab c

^{2}+a^{2 }bcac bc a

^{2}+b^{2}is equal to 4a

^{2}b^{2}c^{2}a

^{2}2ab b^{2}b

^{2}a^{2}2ab = (a^{3}+b^{3})^{2}2ab b

^{2}a^{2}state any short tricks to solve prob. on properties of determinant. and identify how to solve it by slight seeing????????

$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& -2& 4\end{array}\right],I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{A}^{-1}=\frac{1}{6}\left({A}^{2}+CA+dI\right)\phantom{\rule{0ex}{0ex}}Wherec,d\in R,thenpairofvaluesofcanddare.$

Please answer asap

| b^2 +c^2 ab ac |

| ab c^2+a^2 bc |=4a^2b^2c^2

| ca cb a^2+ b^2|

Using properties of determinants, solve the following for x :

x-2 2x-3 3x-4

x-4 2x-9 3x-16 =0

x-8 2x-27 3x-64

prove without expanding that the determinant equals 0

b2c2 bc b-c

c2a2 ca c-a

a2b2 ab a-b

py+z y z

0 px+y py+z

= 0

where p is any real number

|b+c a a |

| b c+a b |=4abc

| c c a+b |

for any 2*2 matrix A, if A(adjA) = [10 0] find A determinant....?

[0 10]

265 240 219

240 225 198

219 198 181

=0

Please solve the following determinant based question | (y+z)^2 xy zx |

| xy (x+z)^2 yz | = 2xyz(x+y+z)^3 .

| xz yz (x+y)^2 |

Please give the answer fast !!

|x1 y1 2 |^2

|x2 y2 2| = 3a^4

|x3 y3 2|

Using properties of determinats, prove that

a

^{2 }2ab b^{2}b

^{2 }a^{2 }2ab2ab b

^{2 }a^{2 }= (a

^{3}+ b^{3})^{2}Without expanding the determinants show that

1 w w2

w w2 1

w2 1 w

=0 where w is one of the cube roots of unity.

(w2 means w square )

px+y x y

py+z y z = 0

0 px+y py+z

prove that a+b+2c a b

c b+c+2a b = 2( a+b+c)

^{3}c a c+a+2b

1. A square matrix A, of order 3, has |A|=5, find |A adj. A|.

What is the formula for Det[ adj( adj(A) ) ] and how do you derive it ?

An amount of Rs. 10,000 is put into three investments at the rate of 10,12 and 15 per cent per annum. The combined income is Rs. 1,310 and the combined income of the first and the second investment is Rs. 190 short of the income from the third.

i) Represent the above situation by matrix equation and form the linear equation using multiplication.

ii) Is it possible to solve the system of equations so obtained using matrices?

Show that the elements along the main diagonal of a skew symmetric matrix are all zero.

Pls. answer

easy way to solve elementary row or column transformation

prove that the 3x3 determinant :

| 1+a

^{2}-b^{2}2ab -2b || 2ab 1-a

^{2}+b^{2}2a | = (1+a^{2}+b^{2})^{3 }| 2b -2a 1-a

^{2}-b^{2}|using the properties of determinants show that..

sin2x cos2x 1

cos2x sin2x 1

-10 12 2

=0..

(sin2x and cos2x means sin square x and cos square x respectively)

= 2(a+b)(b+c)(c+a)

evaluate the following determinants:

matrix of [ad+bc bd-ac

ac-bd ad+bc]

how to solve determinant of 4x4 matrix?

If A is an invertible matrix of order 3 and |A|=5, then find |adj A|

determinant {5

^{2}5^{3}5^{4}5

^{3}5^{4}5^{5}5

^{4}5^{5}5^{6}}find the value of determinantsubscriber. She proposes to increase the annual subscription charges and it is believed that for

every increase of Re 1, one subscriber will discontinue. What increase will bring maximum

income to her? Make appropriate assumptions in order to apply derivatives to reach the

solution. Write one important role of magazines in our lives.

a b-c c+b

a+c b c-a

a-b b+a c =(a+b+c)(a^2+b^2+c^2)

Evaluate the following determinants:

bar of (log

_{a}b 1)(1 log

_{b}a)if A is a square matrix of order 3 such that adj(2A) = k adj(A) , then wite the value of k..

prove that determinant of x x

^{2 }yzy y

^{2}zx = (x-y)(y-z)(z-x)(xy+yz+zx)z z

^{2}xy{1 a2+bc a3

1 b2+ca b3

1 c2+ab c3} = -(a-b) (b-c) (c-a) (a2 +b2+c2) using properties of determinannts solve

A is a square matrix of order 3 and det. A = 7. Write the value of adj A.

Please give me any formula or method for calculating this problem.

To prove :

(b+c)

^{2}a^{2}a^{2}b

^{2}(c+a)^{2}b^{2}= 2abc(a+b+c)^{3}b

^{2}c^{2}(a+b)^{2}(a

^{2}+ b^{2})/c c ca (b

^{2}+ c^{2})/a a = 4abcb b ( c

^{2}+ a^{2})/bWhat is the difference between

+2 ( plus or minus 2 )and minus or plus 2 ?if a,b,c are all positive and are pth,qth,rth terms of a G.P, then show that determinant

|log a p 1|

| log c r 1|

Using the properties of determinants, evaluate:

1

^{2}2^{2}3^{2}2

^{2}3^{2}4^{2}3

^{2}4^{2}5^{2}Solve:

(i) x+y-2z =0 (ii)2x+3y+4z =0 (iii)3x+y+z =0 (iv) x+2y-3z = -4

2x+y-3z =0 x+y+z =0 x-4y+3z =02x+3y+2z =2

5x+4y-9z =0 2x-y+3z =0 2x+5y-2z =0 3x-3y-4z =11

A = [ 2 -3

3 4 ]

satisfies the equation x^2 - 6x + 17 = 0. Hence find A^-1.

If x + y + z = 0, prove that|xa yb zc| |a b c||yc za xb|= xyz |c a b||zb xc ya| |b c a|

Iwant the answer within 2 hours.Please!!!!!!

11. $If{\u25b3}_{r}=\left|\begin{array}{ccc}r-1& n& 6\\ {\left(r-1\right)}^{2}& 2{n}^{2}& 4n-2\\ {\left(r-1\right)}^{3}& 3{n}^{3}& 3{n}^{3}-3n\end{array}\right|,provethat\sum _{r=1}^{n}{\u25b3}_{r}=0.$