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.