# 85. Matrices Problems Part 1

1. Find the number of matrices having $$12$$ elements.

2. Write down the matrix $$A = [a_{ij}]_{2\times 3}$$ where $$a_{ij} = 2i - 3j$$

3. If $$A= \begin{bmatrix}a & b\\-b & a\end{bmatrix}, B=\begin{bmatrix}-a & b\\-b & -a\end{bmatrix},$$ then find $$A + B$$

4. If $$Y = \begin{bmatrix}3 & 2\\1 & 4\end{bmatrix}$$ and $$2X + Y = \begin{bmatrix}1 & 0\\-3 & 2\end{bmatrix},$$ find $$X.$$

5. If $$\begin{bmatrix}x^2 - 4x & x^2\\x^2 & x^3\end{bmatrix} = \begin{bmatrix}-3 & 1\\ x- + 2 & 1\end{bmatrix},$$ then find $$x.$$

6. Find $$x, y, z$$ and $$a$$ for which $$\begin{bmatrix}x + 3 & 2y + x \\ z -1 & 4a - 6\end{bmatrix} = \begin{bmatrix}0 & -7 \\ 3 & 2a\end{bmatrix}$$

7. If $$A = \begin{bmatrix}1 & 2 & 3\\-1 & 0 & 2\\1 & -3 & 1\end{bmatrix}, B = \begin{bmatrix}4 & 5 & 6\\ -1 & 0 & 1\\ 2 & 1 & 2\end{bmatrix},$$ find $$4A - 3B$$

8. If $$A = \begin{bmatrix}1 & -2 & 3 \\ -4 & 2 & 5\end{bmatrix}, B = \begin{bmatrix}2 & 3 \\ 4 & 5 \\ 2 & 1\end{bmatrix},$$ find $$AB$$ and $$BA.$$ Also, show that $$AB\neq BA$$

9. If $$A, B, C$$ are three matrices such that $$A = \begin{bmatrix}x & y & z\end{bmatrix}, B = \begin{bmatrix}a & h & g\\h & b & f \\ g & f & c\end{bmatrix}, C = \begin{bmatrix}x \\ y \\ z\end{bmatrix},$$ then find $$ABC.$$

10. Find the transpose and adjoint of the matrix $$A,$$ where $$A = \begin{bmatrix}1 & 2 & 3\\0 & 5 & 0\\2 & 4 & 3\end{bmatrix}$$

11. Find the inverse of the matrix $$A = \begin{bmatrix}0 & 1 & 2\\1 & 2 & 3\\3 & 1 & 1\end{bmatrix}$$

12. Find the inverse of the matrix $$A= \begin{bmatrix}1 & 2 & 5\\2 & 3 & 1\\-1 & 1 & 1\end{bmatrix}$$ and verify that $$AA^{-1} = 1$$

13. Let $$A = \begin{bmatrix}1 & 2 & 2\\2 & 1 & 2\\2 & 2 & 1\end{bmatrix},$$ prove that $$A^2-4A-5I = 0,$$ hence obtain $$A^{-1}$$

14. Solve the following equations by matrix method:

$$5x + 3y +z = 16\\2x + y + 3z = 19\\x + 2y + 4z = 25$$

15. Find the product of two matrices $$A$$ and $$B$$ where $$A=\begin{bmatrix}-5 & 1 & 3\\7 & 1 & -5\\1 & -1 & 1 \end{bmatrix}, B = \begin{bmatrix}1 & 1 & 2\\ 3 & 2 & 1\\ 2 & 1 & 3\end{bmatrix}$$ and use it for solving the equations

$$x + y + 2z = 1\\3x + 2y + z = 7\\2x + y + 3z = 2$$

16. If $$\begin{bmatrix}x + y & 2 \\ 1 & x - y\end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 1 & 7\end{bmatrix},$$ then find $$x$$ and $$y.$$

17. If $$\begin{bmatrix}x - y & 2x + x_1 \\ 2x - y & 3x + y_1\end{bmatrix} = \begin{bmatrix}-1 & 5 \\ 0 & 13\end{bmatrix}$$ and co-ordinates of points $$P$$ and $$Q$$ be $$(x, y)$$ and $$(x_1, y_1),$$ then find $$PQ.$$

18. Find $$X$$ and $$Y$$ if $$X + Y = \begin{bmatrix}7 & 0 \\ 2 & 5\end{bmatrix}$$ and $$X - Y = \begin{bmatrix}3 & 0 \\ 0 & 3\end{bmatrix}$$

19. Given $$A = \begin{bmatrix} 1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1\end{bmatrix}$$ and $$B = \begin{bmatrix}3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3\end{bmatrix},$$ find the matrix $$C$$ such that $$A + C = B$$

20. If $$A = \begin{bmatrix}2 & 3 & 4 \\ -3 & 0 & 2\end{bmatrix}, B = \begin{bmatrix}3 & -4 & -5 \\ 1 & 2 & 1\end{bmatrix}$$ and $$C = \begin{bmatrix} 5 & -1 & 2 \\ 7 & 0 & 3\end{bmatrix},$$ find the matrix $$X$$ such that $$2A + 3B = X + C$$

21. If $$A = \begin{bmatrix} 1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & 1\end{bmatrix}, B = \begin{bmatrix}4 & 5 & 6 \\ -1 & 0 & 1 \\ 2 & 1 & 2\end{bmatrix}, C = \begin{bmatrix}-1 & 2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2\end{bmatrix},$$ find $$A = 2B + 3C$$

22. If $$P(x) = \begin{bmatrix}\cos x & \sin x \\ -\sin x & \cos x\end{bmatrix},$$ then show that $$P(x).P(y) = P(x + y) = P(y).P(x)$$

23. If $$A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{bmatrix},$$ find $$A^2.$$

24. If $$A=\begin{bmatrix}-1 & 1 & -1 \\ 3 & -3 & 3 \\ 5 & -5 & 5 \end{bmatrix}, B = \begin{bmatrix}0 & 4 & 3 \\ 1 & -3 & -3 \\ -1 & 4 & 4\end{bmatrix},$$ then find $$A^2B^2$$

25. If $$A = \begin{bmatrix}2 & 3 & 4 \\ 1& 2 & 3 \\ -1 & 1 & 2\end{bmatrix}, B = \begin{bmatrix}1 & 3 & 0\\ -1 & 2 & 1 \\ 0 & 0 & 2\end{bmatrix},$$ find $$AB$$ and $$BA$$ and show that $$AB \neq BA$$

26. Find the product of the following two matrices:

$$\begin{bmatrix}0 & c & -b \\ -c & 0 & a \\ b & -a & 0\end{bmatrix}$$ and $$\begin{bmatrix}a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2\end{bmatrix}$$

27. If $$A = \begin{bmatrix}3 & -5 \\ -4 & 2\end{bmatrix},$$ find $$A^2 - 5A - 14I,$$ where $$I$$ is a unit matrix.

28. Verify that $$A = \begin{bmatrix}2 & 3\\ 1 & 2\end{bmatrix}$$ satisfies the equation $$A^3 - 4A^2 + A = O$$

29. If $$A = \begin{bmatrix}0.8 & 0.6 \\ -0.6 & 0.8\end{bmatrix},$$ find $$A^2$$

30. If $$A = \begin{bmatrix}3 & 1 \\ -1 & 2\end{bmatrix},$$ find $$f(A),$$ where $$f(x) = x^2 - 5x + 7I$$

31. If $$A=\begin{bmatrix}\cos\theta & \sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}, B = \begin{bmatrix}\cos\phi & \sin\phi \\ \sin\phi & \cos\phi \end{bmatrix},$$ show that $$AB = BA$$

32. Let $$f(x) = x^2 - 5x + 6,$$ find $$f(A),$$ if $$A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 9\end{bmatrix}$$

33. If the matrix $$A = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix},$$ then verify that $$A^2 - 12 A - I = 0,$$ where $$I is a unit matrix.$$

34. Show that $$\begin{pmatrix}\begin{bmatrix}1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{bmatrix} + \begin{bmatrix} \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \\ \omega & \omega^2 & 1\end{bmatrix}\end{pmatrix} \begin{bmatrix}1 \\ \omega \\ \omega^2 \end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}$$

35. Let $$A = \begin{bmatrix}0 & -\tan\frac{\alpha}{2} \\ \tan\frac{\alpha}{2} & 0\end{bmatrix}$$ and $$I,$$ the identity matrix of order $$2.$$ Show that $$I+ A = (I - A) \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha\end{bmatrix}$$

36. Without using the concept of inverse of matrix, find the matrix $$\begin{bmatrix} x & y \\ z & u\end{bmatrix}$$ such that $$\begin{bmatrix} 5 & -7 \\ -2 & 3\end{bmatrix} \begin{bmatrix} x & y \\ z & u \end{bmatrix} = \begin{bmatrix} -16 & -6 \\ 7 & 2\end{bmatrix}$$

37. Find $$x$$ so that $$\begin{bmatrix}1 & x & 1\end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 3 & 2 \end{bmatrix} \begin{bmatrix}1 \\ 1 \\ x\end{bmatrix} = O$$

38. Prove that the product of two matrices $$\begin{bmatrix} \cos^2\theta & \cos\theta\sin\theta \\ \cos\theta\sin\theta & \sin^2\theta\end{bmatrix}$$ and $$\begin{bmatrix} \cos^2\phi & \cos\phi\sin\phi \\ \cos\phi\sin\phi & \sin^2\phi\end{bmatrix}$$ is a zero matrix when $$\theta$$ and $$\phi$$ differ by an odd multiple of $$\frac{\pi}{2}$$

39. If $$A = \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix},$$ then show that $$A^n = \begin{bmatrix}\cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta\end{bmatrix}$$ where $$n$$ is a positive integer.

40. If $$A = \begin{bmatrix}3 & -4 \\ 1 & -1\end{bmatrix},$$ show that $$A^n = \begin{bmatrix}1 + 2n & -4n \\ n & 1 - 2n\end{bmatrix},$$ where $$n$$ is a positive integer.

41. Let $$A = \begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix}.$$ Show hat $$(aI + bA)^n = a^nI + na^{n - 1}bA,$$ where $$I$$ is a unit matrix of order $$2$$ and $$n$$ is a positive integer.

42. Under what condition is the marix equation $$A^2 - B^2 = (A + B)(A - B)$$ true?

43. A man buys $$8$$ dozens of mangoes, $$10$$ dozens of apples and $$4$$ dozens of bananas.Mangoes cost USD $$18$$ per dozen, apples $$9$$ per dozen and bananas $$6$$ per dozen. Represent the quntities by a row and a column matrix. Also, find the total cost.

44. A trust fund has USD $$30,000$$ that is to be invested in two different types of bonds. The first bond pays $$5%$$ interest per year and second bond pays $$7%$$ interest per year. using matrix multiplication determine how to divide USD $$30,000$$ among the two types of bonds if the turst find must obtain an annual interest of USD $$2000.$$

45. A store has in stock $$20$$ dozen shirts, $$15$$ dozen trousers and $$25$$ dozen pair of socks. If the selling prices are USD $$50$$ per shirt, $$90$$ per trouser and $$12$$ per pair of socks, then find the toal amount store owner will get after selling all the items in the stock.

46. Co-operative store of a particular school has $$10$$ dozen physics books, $$8$$ dozen chemisty books and $$5$$ dozen mathematics books. Their selling prices are USD $$8.3, 3.45, 4.5$$ each respectively. Find the total amnount the store owner will receive after selling all the books.

47. If $$A = \begin{bmatrix}\cos\alpha & \sin\alpha \\ -\sin\alpha & \cos \alpha\end{bmatrix},$$ verify that $$AA' = I_2 = A'A$$

48. Express the following matrix as a sum of a symmetric matrix and skew symmetric matrix $$\begin{bmatrix}1 & 2 & 4 \\ 6 & 8 & 1 \\ 3 & 5 & 7\end{bmatrix}$$

49. Show that the following matrix is orthogonal $$\begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha\end{bmatrix}$$

50. Show that the matrix $$\frac{1}{3}\begin{vmatrix} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1\end{vmatrix}$$ is orthogonal.