# 76. Determinants Problems Part 1

1. Evaluate $$\begin{vmatrix}4 & 9 & 7\\ 3 & 5 & 7\\5 & 4& 5\end{vmatrix}$$

2. Show that $$\begin{vmatrix}1 & a & a^2\\1 & b & b^2\\1 & c & c^2\end{vmatrix} = (a -b)(b - c)(c - a)$$

3. Evaluate $$\begin{vmatrix}1 & 2 & 4\\1 & 3 & 9\\1 & 4 & 16\end{vmatrix}$$ making use of relation between 2nd and 3rd column.

4. Let $$a, b, c$$ be positive and unequal. Show that the value of the determinant $$\begin{vmatrix}a & b & c\\b & c & a \\c & a & b\end{vmatrix}$$ is negative.

5. Evaluate $$\begin{vmatrix}b + c & a + b & a\\c + a & b + c & b\\a + b & c + a & c\end{vmatrix}$$

6. Evaluate $$\begin{vmatrix}1 + a_1 & a_2 & a_3\\a_1 & 1 + a_2 & a_3\\a_1 & a_2 & 1 + a_3\end{vmatrix}$$

7. Show that $$\begin{vmatrix}a + b + 2c & a & b\\c & b + c + 2a & b\\c & a & c + a + 2b\end{vmatrix} = 2(a + b + c)^3$$

8. Show that $$\begin{vmatrix}a - b + c & a + b - c & a - b - c\\b - c + a& b + c -a & b - c - a\\c - a + b & c + a - b & c - a -b\end{vmatrix} = 4(a^3 + b^3 + c^3 - 3abc)$$

9. Prove that $$\begin{vmatrix}a - b - c & 2a & 2a\\2b & b - c -a & 2b\\2c & 2c & c - a -b\end{vmatrix} = (a + b + c)^3$$

10. Prove that $$\begin{vmatrix}x & y & z\\x^2 & y^2 & z^2\\yz & zx & xy\end{vmatrix} = \begin{vmatrix}1 & 1 & 1\\ x^2 & y^2 & z^2\\ x^3 & y^3 & z^3\end{vmatrix} = (y - z)(z - x)(x - y)(xy + yz + zx)$$

11. Prove that $$\begin{vmatrix}a^2 + 1 & ab & ac\\ab & b^2 + 1 & bc\\ac & bc & c^2 + 1\end{vmatrix} = 1 + a^2 + b^2 + c^2$$

12. Prove that $$\begin{vmatrix}1 + a_1 & 1 & 1\\1 & 1 + a_2 & 1\\1 & 1 & 1 + a_3\end{vmatrix} = a_1a_2a_3\left(1 + \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3}\right)$$

13. If $$x, y, z$$ are all different and if $$\begin{vmatrix}x & x^2 & 1 + x^3\\y & y^2 & 1 + y^3\\z & z^2 & 1+z^3\end{vmatrix} = 0,$$ prove that $$xyz = -1$$

14. Evaluate $$\begin{vmatrix}b + c & a & a\\b & c + a & b\\c & c & a + b\end{vmatrix}$$

15. Show that $$\begin{vmatrix}(b + c)^2 & a^2 & a^2\\b^2 & (c + a)^2 & b^2\\c^2 & c^2 & (a + b)^2\end{vmatrix} = 2abc(a + b + c)^3$$

16. Solve the equation $$\begin{vmatrix}15 -x & 1 & 10\\11 - 3x& 1 & 16\\7 - x & 1 & 13\end{vmatrix} = 0$$

17. If $$a + b + c = 0,$$ solve the equation $$\begin{vmatrix}a - x & c & b\\c & b - x & a\\b & a & c - x\end{vmatrix} = 0$$

18. If $$D_1 = \begin{vmatrix}a & b & c\\d & e & f\\g & h &k\end{vmatrix}, D_2 = \begin{vmatrix}a & g & x\\b & h & y\\c & k & z\end{vmatrix}$$ and $$d = tx, e = ty, f= tz,$$ prove without expanding that $$D_1 = -tD_2.$$

19. Show without expanding that $$\begin{vmatrix}a & bc & abc\\b & ca & abc\\c & ab & abc\end{vmatrix} = \begin{vmatrix}a & a^2 & a^3\\b & b^2 & b^3\\c & c^2 & c^3\end{vmatrix}$$

20. If $$a, b, c$$ are positive and are the $$p\text{th}, q\text{th}, r\text{th}$$ terms of a G.P., respectively, then show without expanding that $$\begin{vmatrix}\log a & p & 1\\\log b & q & 1\\\log c & r & 1\end{vmatrix} = 0$$

21. Evaluate $$\begin{vmatrix}4 & 9 & 2\\3 & 5 & 7\\8 & 1 & 6\end{vmatrix}$$

22. Evaluate $$\begin{vmatrix}18 & 1 & 17\\22 & 3 & 19\\26 & 5 & 21\end{vmatrix}$$

23. Evaluate $$\begin{vmatrix}4 & 9 & 7\\3 & 5 & 7\\5 & 4 & 5\end{vmatrix}$$

24. Evaluate $$\begin{vmatrix}1^2 & 2^2 & 3^2\\2^2 & 3^2 & 4^2\\3^2 & 4^2 & 5^2\end{vmatrix}$$

25. Evaluate $$\begin{vmatrix}1 & 1 & 1\\1 & 1 + x & 1\\1 & 1 & 1 + y\end{vmatrix}$$

26. Evaluate $$\begin{vmatrix}1 & 1 & 1\\a & b & c\\a^3 & b^3 & c^3\end{vmatrix}$$

27. Evaluate $$\begin{vmatrix}1 & b + c & b^2 + c^2\\1 & c + a & c^2 + a^2\\ 1 & a + b & a^2 + b^2\end{vmatrix}$$

28. Evaluate $$\begin{vmatrix}1 & a & a^2 - bc\\1 & b & b^2 - ca \\ 1 & c & c^2 - ab\end{vmatrix}$$

29. Evaluate $$\begin{vmatrix}1 & bc & bc(b + c)\\1 & ca & ca(c + a)\\1 & ab & ab(a + b)\end{vmatrix}$$

30. Prove that $$\begin{vmatrix}1 & a & b + c\\1 & b & c + a\\1 & c & a + b\end{vmatrix} = 0$$

31. If $$a, b, c$$ be the $$p\text{th}, q\text{th}, r\text{th}$$ terms respectively of an H.P., show that $$\begin{vmatrix}bc & p & 1\\ ca & q & 1\\ ab & r & 1\end{vmatrix} = 0$$

32. If $$\begin{vmatrix}x^2 + 3x & x - 1 & x + 3\\x +1 & 1 -2x & x - 4\\ x- 2 & x + 4& 3x\end{vmatrix} = px^4 + qx^3 + rx^2 + sx + t$$ be an identity in $$x,$$ where $$p, q, r, s$$ and $$t$$ are constants, find the value of $$t$$.

33. Prove that $$\begin{vmatrix}a & b & c\\a^2 & b^2 & c^2\\a^3 & b^3 & c^3\end{vmatrix} = abc(a - b)(b - c)(c - a)$$

34. If $$a, b, c$$ are in A. P., show that $$\begin{vmatrix}x + 1 & x + 2 & x + a\\ x + 2& x+ 3& x + b\\ x+3 & x + 4 & x + c\end{vmatrix} = 0$$

35. If $$\omega$$ is a complex (non real) cube root of unity, prove that $$\begin{vmatrix}1 & \omega & \omega^2\\\omega & \omega^2 & 1\\\omega & 1 & \omega^2\end{vmatrix} = 0$$

36. Evaluate $$\begin{vmatrix}k & k & k\\1 & 2 & 3\\1 & 3 & 6\end{vmatrix}$$

37. Evaluate $$\begin{vmatrix}a^2 + x & b^2 & c^2\\a^2 & b^2 + x + c^2\\a^2 & b^2 & c^2 + x\end{vmatrix}$$

38. Evaluate $$\begin{vmatrix}a & b + c & a^2\\b & c + a & b^2\\c & a + b & c^2\end{vmatrix}$$

39. Evaluate $$\begin{vmatrix}b + c& a - b & a\\c + a & b - c & b\\a + b & c - a & c\end{vmatrix}$$

40. Show that $$\begin{vmatrix}a + b & b + c & c + a\\b + c & c + a & a + b\\c + a & a + b & b + c\end{vmatrix} = -2(a^3 + b^3 + c^3 - 3abc)$$

41. Show that $$\begin{vmatrix}x + a & x + b & x + c\\y + a & y + b & y + c\\z + a & z + b & z + c\end{vmatrix} = 0$$

42. Show that $$\begin{vmatrix}0 & p - q & p - r\\q - p & 0 & q -r\\r -p & r - q & 0\end{vmatrix} = 0$$

43. Show that $$\begin{vmatrix}a & a + b & a + 2b\\a + 2b & a & a + b\\a + b & a + 2b & a\end{vmatrix} = 9b^2(a + b)$$

44. Show that $$\begin{vmatrix}a & b - c & c + b\\a + c & b & c- a\\a - b & b + a & c\end{vmatrix}$$ and $$(a + b + c)$$ have same sign.

45. Evaluate $$\begin{vmatrix}b^2 + c^2 & ab & ac\\ab & c^2 + a^2 & bc\\ca & cb & a^2 +b^2\end{vmatrix}$$

46. Show that $$\begin{vmatrix}(b + c)^2 & c^2 & b^2\\c^2 & (c + a)^2 & a^2\\b^2 & a^2 & (a + b)^2\end{vmatrix} = 2(bc + ca + ab)^3$$

47. Show that $$\begin{vmatrix}(a + b)^2 & ca & bc\\ca & (b + c)^2 & ab\\bc & ab & (c + a)^2\end{vmatrix} = 2abc(a + b + c)^3$$

48. Show that $$\begin{vmatrix}\frac{a^2 + b^2}{c} & c & c\\a & \frac{b^2 + c^2}{a} & a\\b & b & \frac{c^2 + a^2}{b}\end{vmatrix} = 4abc$$

Solve the following equations:

1. $$\begin{vmatrix}a & a & x\\a & a & a\\b & x & b\end{vmatrix} = 0$$

2. $$\begin{vmatrix}x & 2 & 3\\6 & x + 4 & 4\\7 & 8 & x + 8\end{vmatrix} = 0$$