# 75. Determinants Problems Part 4¶

1. For positive numbers $$x, y , z$$ (not equal to $$1$$), show that the numerical value of the determinant

$$\begin{vmatrix}1 & \log_xy & \log_xz \\ \log_yx & 1 & \log_yz \\ \log_zx & \log_zy & 1\end{vmatrix} = 0$$

2. If $$a, b, c > 0$$ and $$x, y , z \in R,$$ then show without expanding that

$$\begin{vmatrix}(a^x + a^{-x})^2 & (a^x - a^{-x})^2 & 1 \\ (b^y + b^{-y})^2 & (b^y - b^{-y})^2 & 1 \\ (c^z + c^{-z})^2 & (c^z - c^{-z})^2 & 1\end{vmatrix} = 0$$

3. Wihtout expanding the determinants prove that

$$\begin{vmatrix}103 & 115 & 114 \\ 111 & 108 & 106 \\ 104 & 113 & 116\end{vmatrix} + \begin{vmatrix}113 & 116 & 104 \\ 108 & 106 & 111 \\ 115 & 114 & 103\end{vmatrix} = 0$$

4. Evaluate $$\sum_{n = 1}^N U_n$$ if $$U_n = \begin{vmatrix}n & 1 & 5 \\ n^2 & 2N + 1 & 2N+ 1 \\ n^3 & 3N^2 & 3N\end{vmatrix}$$

5. If $$A, B, C$$ are angles of a triangle, then show without expanding that

$$\begin{vmatrix}\sin(A + B + C) & \sin B & \cos C \\ -\sin B & 0 & \tan A \\ \cos (A + B) & -\tan A & 0\end{vmatrix} = 0$$

6. Evaluate without expanding $$\begin{vmatrix}b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 -a b \\ bc - ac & c - a & ab - a^2\end{vmatrix}$$

7. Let $$\Delta_i = \begin{vmatrix}i - 1 & n & 6 \\ (i - 1)^2 & 2n^2 & 4n - 2 \\ (i - 1)^3 & 3n^3 & 3n^2 - 2n\end{vmatrix}$$

Show that $$\sum_{i = 1}^n \Delta_i = k,$$ a constant.

8. Let $$m$$ be a positive integer and $$\Delta_r = \begin{vmatrix}2r - 1 & {}^mC_r & 1 \\ m^2 - 1 & 2^m & m + 1 \\ \sin^2m^2\ & \sin^2 m & \sin^2(m + 1) \end{vmatrix}$$

Then find the value of $$\sum_{r = 0}^m\Delta_r$$

9. Show that $$\begin{vmatrix}{}^xC_r & {}^xC_{r + 1} & {}^xC_{r + 2} \\ {}^yC_r & {}^yC_{r + 1} & {}^yC_{r + 2} \\ {}^zC_r & {}^zC_{r + 1} & {}^zC_{r + 2} \end{vmatrix} = \begin{vmatrix}{}^xC_r & {}^{x + 1}C_{r + 1} & {}^{x + 2}C_{r + 2} \\ {}^yC_r & {}^{y + 1}C_{r + 1} & {}^{y + 2}C_{r + 2} \\ {}^zC_r & {}^{z + 1}C_{r + 1} & {}^{z + 2}C_{r + 2}\end{vmatrix}$$

10. If $$\Delta_r = \begin{vmatrix}r & n + 1 & 1 \\ r^2 & 2n - 1 & \frac{2n + 1}{3} \\ r^3 & 3n + 2 & \frac{n(n + 1)}{2}\end{vmatrix},$$

show that $$\sum_{r = 1}^n \Delta_r = 0$$

11. If $$\Delta_r = \begin{vmatrix}2^{r - 1} & 2.3^{r - 1} & 4.5^{r - 1} \\ x & y & z \\ 2^{n} - 1 & 3^n - 1 & 5^n - 1\end{vmatrix},$$

show that $$\sum_{r = 1}^n \Delta_r = 0$$

12. Show without expanding that $$\begin{vmatrix} x^2 & (x - 1)^2 & (x - 2)^2 \\ (x - 1)^2 & (x - 2)^2 & (x - 3)^2 \\ (x - 2)^2 & (x - 3)^2 & (x - 4)^2\end{vmatrix}$$ is independent of $$x.$$

13. Show without expanding that $$\begin{vmatrix}2 & 1 + i & 3 \\ 1 - i & 0 & 2 + i \\ 3 & 2 - i & 1\end{vmatrix}$$ is purely real.

14. Show without expanding that $$\begin{vmatrix}x - 3 & 2x +1 & 2 \\ 3x + 2 & x + 2 & 1 \\ 5x + 1 & 5x + 4 & 5\end{vmatrix}$$ is independent of $$x.$$

15. If $$a$$ and $$x$$ are real numbers and $$n$$ is a postive integer, then show without expanding that

$$\begin{vmatrix}a^n - x & a^{n + 1} - x & a^{n + 2} - x \\ a^{n + 3} - x & a^{n + 4} - x & a^{n + 5} - x \\ a^{n + 6} - x & a^{n + 7} - x & a^{n + 8} - x\end{vmatrix} = 0$$

16. Find $$\sum_{r = 2}^n (-2)^r \begin{vmatrix}{}^{n - 2}C_{r - 2} & {}^{n - 2}C_{r - 1} & {}^{n - 2}C_r \\ -3 & 1 & 1 \\ 2 & 1 & 0\end{vmatrix}, n > 2$$

17. If $$a,b,c$$ are non-zero real numbers, show without expanding that

$$\begin{vmatrix}b^2c^2 & bc & b + c \\ c^2a^2 & ca & c + a \\ a^2b^2 & ab & a + b\end{vmatrix} = 0$$

18. Prove that $$\begin{vmatrix}b + c -a - d & bc - ad & bc(a + d) - ad(b + c) \\ c + a - b - d & ca - bd & ca(b + d) - bd(c + a) \\ a + b - c - d & ab - cd & ab(c + d) - cd(a + b)\end{vmatrix} = \\-2(b - c)(c - a)(a - b)(a - d)(b - d)(c - d)$$

19. Prove that $$\begin{vmatrix}bc - a^2 & ca - b^2 & ba - c^2 \\ ca + ab - bc & bc + ab -ca & bc + ca - ab \\ (a + b)(a + c) & (b + c)(b + a) & (c + a)(c + b)\end{vmatrix} = \\3(b - c)(c - a)(a - b)(a + b + c)(ab + bc + ca)$$

20. Prove that $$\begin{vmatrix}1 & (m + n - l - p)^2 & (m + n - l - p)^2 \\ 1 & (n + l -m - p)^2 & (n + l -m - p)^4 \\ 1 & (l + m - n - p)^2 & (l + m - n - p)^4\end{vmatrix} = \\ 64(l - m)(l - n)(l - p)(m - n)(m - p)(n - p)$$

21. If $$u, v, w$$ are differentiable functions of $$f$$ and suffixes denote the derivatives w.r.t. $$t$$, prove that

$$\frac{d}{dt}\begin{vmatrix}u_1 & v_1 & w_1 \\ u_2 & v_2 & w_2 \\ u_3 & v_3 & w_3\end{vmatrix} = \begin{vmatrix}u_1 & v_1 & w_1 \\ u_2 & v_2 & w_2 \\ u_4 & v_4 & w_4\end{vmatrix}$$

22. If $$Y= sX$$ and $$Z = tX,$$ all the variables being differentiable functions of $$x,$$ prove that

$$\begin{vmatrix}X & Y & Z \\ X_1 & Y_1 & Z_1 \\ X_2 & Y_2 & Z_2\end{vmatrix} = X^3\begin{vmatrix}s_1 & t_1 \\ s_2 & t_2\end{vmatrix}$$ where suffixes denote the derivative w.r.t. $$x.$$

23. If $$f(x), g(x), h(x)$$ are polynomials in $$x$$ find the condition that

$$\begin{vmatrix}f(x) & g(x) & h(x) \\ f(\alpha) & g(\alpha) & h(\alpha) \\ f(\beta) & g(\beta) & h(\beta)\end{vmatrix}$$

which a polynomial of degree $$3,$$ is exressible as $$a(x - \alpha)^2(x - \beta)$$

24. Show that $$\begin{vmatrix}\sin(x + \alpha) & \cos(x + \alpha) & a + x\sin\alpha \\ \sin(x + \beta) & \cos(x + \beta) & b + x\sin\beta \\ \sin(x + \gamma) & \cos(x + \gamma) & c + x\sin\gamma\end{vmatrix}$$ is independent of $$x.$$

25. If $$f(x) = \begin{vmatrix}2\cos^2x & \sin 2x & -\sin x \\ \sin 2x & 2\sin^2 x & \cos x \\ \sin x & -\cos x & 0\end{vmatrix},$$ show that $$\int_{0}^{\frac{\pi}{2}} (f(x) + f'(x))dx = \pi$$

26. Prove that $$\begin{vmatrix}a_1\alpha_1 + b_1\beta_1 & a_1\alpha_2 + b_1\beta_2 & a_1\alpha_3 + b_1\beta_3 \\ a_2\alpha_1 + b_2\beta_1 & a_2\alpha_2 + b_2\beta_2 & a_2\alpha_3 + b_2\beta_3 \\ a_3\alpha_1 + b_3\beta_1 & a_3\alpha_2 + b_3\beta_2 & a_3\alpha_3 + b_3\beta_3\end{vmatrix} = 0$$

27. If $$l_r\vec{i}, m_r\vec{j}, n_r\vec{k}, r = 1,2,3$$ be three mutually perpendicular unit vectors, show that

$$\begin{vmatrix}l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \\ n_1 & n_2 & n_3\end{vmatrix} = \pm 1$$

28. Let $$\Delta = \begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_13\end{vmatrix}$$ and $$A_i, B_i, C_i$$ be the cofactors of $$a_i, b_i, c_i$$ respectively and $$\alpha_i, \beta_i, \gamma_i$$ be the cofactors of $$A_i, B_i, C_i$$ respectively, where $$i = 1,2,3,$$ show that

$$\begin{vmatrix}A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3\end{vmatrix} \begin{vmatrix}\alpha_1 & \beta_1 & \gamma_1 \\ \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_3 & \beta_3 & \gamma_3 \end{vmatrix} = \Delta^6$$

29. Using determinants solve the equations:

$$x + 2y + 3z = 6; 2x + 4y + z = 17; 3x + 2y + 9z = 2$$

30. Solve the system of equations:

$$ax + by + ca = d, a^2x + b^2y + c^2a = d^2; a^3x + b^3y + c^3a = d^3$$

Will the solution always exist and be unique?

31. Determine the coefficients $$a, b, c$$ of the quadratic function $$f(x)$$ where $$f(x) = ax^2 + bc + c$$ if $$f(1) = 0, f(2) = -2, f(3) = -6$$

32. Determine the coefficients $$a, b, c$$ of the quadratic function $$f(x)$$ where $$f(x) = ax^2 + bc + c$$ if $$f(0) = 6, f(2) = 11, f(-3) = 6$$ also find $$f(1).$$

33. Solve

$$\begin{array} (b + c)(y + z) - ax = b - c \\ (c + a)(z + x) - by = c - a \\ (a + b)(x + y) - cz = a - b\end{array}$$

where $$a + b + c \neq 0$$

34. Examine the consistency of the system of equations

$$7x - 7y + 5z = 3; 3x + y + 5z = 7; 2x + 3y + 5z = 5$$

35. Find the value of $$k$$ for which the following system of equations are consistent

$$x + y = 3; (1 + k)x + (2 + k)y = 8; x - (1 + k)y + (2 + k) = 0$$

36. Find the value of $$k$$ for which the following system of equations are consistent

$$(k + 1)^3x + (k + 2)^3y = (k + 1)^3; (k + 1)x + (k + 2)y = k + 3; x + y = 1$$

37. Find the values of $$c$$ for which the equations

$$2x + 3y = 3; (c + 2)x + (c + 4)y = c + 6; (c + 2)^2x + (c + 4)^2y = (c + 6)^2$$

38. Find $$\lambda$$ for which the system of equations

$$x + y - 2z = 0; 2x - 3y + z = 0; x - 5y + 4z = \lambda$$ are consistent and find the solutions for all such values of $$\lambda$$

39. Find the value of $$\lambda$$ and $$\mu$$ for which the following system of equations

$$x + y + z = 0; x + 2y + 3z = 14; 2x + 5y + \lambda z = \mu, \lambda, \mu \in R$$ has (i) unique solution (ii) infinite solutions.

40. If $$bc + qr = ca + rp = ab + pq = -1,$$ show that

$$\begin{vmatrix}ap & a & p \\ bq & b & q \\ cr & c & r\end{vmatrix} = 0$$

41. Find all values of $$k$$ for which the following system posses a non-trivial solution:

$$x + ky + 3z = 0; kx + 2y + 2z = 0; 2x + 3y + 4z = 0$$

42. If $$x = cy + bz; y = az + cx; z = bx + ay$$ where $$x,y,z$$ are not all zero. Prove that $$a^2 + b^2 + c^2 + 2abc = 1.$$

Further if at least one of $$a,b,c$$ is a proper fraction, prove that

1. $$a^2 + b^2 + c^2 < 3$$

2. $$abc > -1$$

43. If $$a = \frac{x}{y - z}, b = \frac{y}{z - x}, c = \frac{z}{x - y}$$ where $$x,y,z$$ are not all zero, prove that $$1 + ab + bc + ca = 0$$

44. Consider the system of linear equations, in $$x,y,z$$:

$$\begin{array}((\sin 3\theta)x - y + z = 0\\ (\cos 2\theta)x + 4y + 3z = 0 \\ 2x + 7y + 7z = 0\end{array}$$

Find the value of $$\theta$$ for which this system has non-trivial solution.

45. If $$a,b,c$$ are in G.P. with common ration $$r_1$$alpha, beta, gamma` are in G.P. with common ratio $$r_2,$$ then find the conditions that $$r_1$$ must satisfy in order that the equations

$$ax + \alpha y + z = 0; bx + \beta y + z = 0; cx + \gamma y + z = 0$$

have only trivial solutions.