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Show that: |x p q p x q q q x|=(x-p)(x^2...

Show that: `|x p q p x q q q x|=(x-p)(x^2+p x-2q^2)dot`

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Given, `|[x, p, q],[ p, x, q],[ q, q, x]|`
Applying `C_1​->C_1​−C_2`
we have
`|[x-p, p-x, 0],[ p, x, q],[ q, q, x]|`
=`(x-p)|[1, -1, 0],[ p, x, q],[ q, q, x]|`
Applying `R_1​->R_1​+R_2`
​=`(x-p)|[0, p+x, 2p],[ p, x, q],[ q, q, x]|`
Expanding along `C_1` we have
...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Without expanding or evaluating show that |[0 , b-a , c-a] , [a-b , 0...

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  2. If A is a skew-symmetric matrix of odd order n , then |A|=0

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  3. Show that: |x p q p x q q q x|=(x-p)(x^2+p x-2q^2)dot

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  4. If f(x)=|(a,-1 ,0),(ax,a,-1),(a x^2,a x, a)|, find the value of f(2x)-...

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  5. If a , b , c are distinct real numbers and the system of equations a x...

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  6. If x , y , z are not all zero such that a x+y+z=0, x+b y+z=0, ...

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  7. Find the value of lambda for which the homogeneous system of equations...

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  8. If the system of equations x=c y+b z y=a z+c x z=b x+a y has...

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  9. A matrix A of order 3xx3 has determinant 5. What is the value of |3A|?

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  10. If A is a square matrix such that |A|=2 , write the value of |AA^T|

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  11. Find the real values of lambda for which the following system of linea...

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  12. If a , b , c are non-zero real numbers and if the system of equations...

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  13. Which of the following is not correct in a given determinant of A , wh...

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  14. Let |(x,2,x),(x^2,x,6),(x,x,6)|=a x^4+b x^3+c x^2+dx+edot Then, the va...

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  15. Let △=|(A x,x^2, 1),(B y, y^2, 1),(C z, z^2, 1)| and△1=|(A,B, C),( x, ...

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  16. If Delta1=|[a, b, c],[ x, y, z],[ p, q, r]| and Delta2=|[q,-b, y],[-p,...

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  17. Without expanding show that: =|cos e c^2thetacot^2theta1cot^2thetacos ...

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  18. Find the value of the determinant =|2 3 4 5 6 8 6x9x 12 x|dot

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  19. Without expanding evaluate the determinant |[41,1,5],[79,7,9],[29,5,3]...

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  20. Show that matrix |(1,a,b+c),(1,b,c+a),(1,c, a+b)|=0

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