CBSE Important Questions

CBSE Guess > Papers > Important Questions > Class XII > 2007 > Maths > Matrices Determinants By Mr. Ashok Kumar

Matrices Determinants

This is the easiest topic for students .All you need is practice .It contributes to about 12 marks to the whole syllabus.
Following formulae may be used:-

Properties of determinants :-

  1. The value of a determinant remains unchanged if its rows and columns are interchanged.
  2. The sign of value of a determinant is changed if its any two rows or columns are interchanged.
  3. The value of a determinant is zero if its any two rows or columns are identical.
  4. If a determinant is multiplied by a scalar (number) , its only one row or column gets multiplied by that constant.
  5. If any two row or column of a determinant are proportional, its value becomes zero.
  6. If all elements of a row or column are expressed as sum of two or more elements ,the whole of the determinant can be expressed in sum of two or more determinants.
  7. If some multiple of one row or column is added or subtracted to another row or column (elementwise), its value remains unchanged.

Tips to solve properties based problems:-

  1. If a determinant is of order ,we can apply only n-1 propertis at a time to it.
  2. The format of application of properties is :-
    Row affected Row affected n (Row used)
    Ex.
  3. The format for interchanging Rows or columns :-
  4. You can never multiply a number to Row affected, it is always multiplied to Row used.
  5. First always try to make elements of any one row or column identical
    so that you could take out common from that row or column. It
    makes all the elements of that Row or column unity(1) and then you
    make at the most two elements of that row or columns zero (0).Now
    expand that the determinant by that row or column.

    Example -

    We shall apply

    and

    Taking out common b from

    Expand by =

  6. Check the part which is required to prove ,try to take out common the factors which are given in the part.

Example -

Here the first factor is (a-b) ,we can obtain it by . Another factor is b-c which we can obtain by

Important Questions are:-

1. Find x,y,z if

[x 3 2] [0 0 1]

2. If A = Show that and hence find .

3. Using properties of determinants solve for x:

4. Using properties prove that :

5. If , prove that ,n N

6. Usining properties of determinants,prove that:

7. Using properties ,prove that:

8. If A = ,Using principle of mathematically induction prove that

9. Using properties of determinants,prove that = 0.

10. If A = , B = , c = Find a matrix D such that CD - AB = 0.

11. Let A = , Verify that.

12. If A = ,find k so that.

13. Find X and Y if.

14. If

Show that

15.Find B if .

16. Find A = , find a and b such that such that where I is unit matrix of order 2.

17. Express as a sum of a symmetric and a skew – symmetric matrix.

18. Prove using properties of determinants.

19. Solve the equations by matrix method

20. If find A-1and use it solve the system of equations:

21. Using determinants ,solve the following system of equations:

22. Find the value of for which given homogeneous system of equations have non trival solution. Also find the solution.

23. If A = find A-1.
UsingA-1solve the system of linear equations:

24. If A = and B = find the product AB and use this result to solve the following system of equations:

25. Solve using matrices :

26. For what value of a and b ,the following system of equations is consistent?