# SOLVED PROBLEMS ON DETERMINANTS

Problem 1 :

Without expanding the determinant, prove that

Solution

Problem 2 :

Show that

Solution

Problem 3 :

Prove that

Solution

Problem 4 :

Prove that

Solution

Problem 5 :

Prove that

Solution

Problem 6 :

Show that

Solution

Problem 7 :

Write the general form of a 3 × 3 skew-symmetric matrix and prove that its determinant is 0.    Solution

Problem 8 :

If

prove that a, b, c are in G.P. or a is a root of ax2 + 2bx + c = 0.             Solution

Problem 9 :

prove that

Solution

Problem 10 :

If a, b, c are pth, qth and rth terms of an A.P, find the value of

Solution

Problem 11:

Prove that

Solution

Problem 12 :

If a, b, c are all positive, and are pth, qth and rth terms of a G.P., show that

Solution

Problem 13 :

Find the value of

if x, y and z ≠ 1       Solution

Problem 14 :

If A =

Solution

Problem 15 :

Without expanding, evaluate the following determinants

Solution

Problem 16 :

If A is a square matrix and | A | = 2, find the value of |AAT| .                 Solution

Problem 17 :

If A and B are square matrices of order 3 such that | A | = -1 and |B| = 3, find the value of |3AB|.      Solution

Problem 18 :

If λ = - 2, determine the value of

Solution

Problem 19 :

Determine the roots of the equation

Solution

Problem 20 :

Verify that det(AB) = (det A) (det B) for

Solution

Problem 21 :

Using cofactors of elements of second row, evaluate | A |, where

Solution

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