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100 Super Revision Test — CBSE Class 12 Matrices

100 Super Revision Test — CBSE Class 12 Matrices - Coding Bihar
100 Super Revision Test — CBSE Class 12 Matrices

🔥 The Night Before the Exam – A Student Preparing for Board Exam 

It was 11:47 PM. Aarav sat at his study table, a cup of cold coffee by his side, pages of Maths formulas and derivations scattered like puzzle pieces. His CBSE Class 12 board exam was just a week away. Panic? A little. Determination? A lot.

“If only I had something that covers everything in one place,” he sighed.

That’s exactly how this 100 Super Revision Test was born — a single collection of the most important questions from every corner of your syllabus. Designed not just to test you, but to prepare you like a topper.

💡 What You’ll Get Here

  • Chapter-wise exam-focused questions to strengthen your concepts.
  • Step-by-step solutions and final answers (for quick last-minute checks).
  • Mix of MCQs, short answers, and numericals — just like your real exam pattern.

So grab your notebook, a cup of chai or coffee, and let’s dive into your 100 Super Revision Challenge that can turn your preparation from stressful to powerful!

🚀 100 Super Revision Test — CBSE Class 12 Matrices (2026)

  1. Find the determinant of \( \begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix} \).
  2. Find the inverse of \( \begin{pmatrix}2 & 1\\ 1 & 1\end{pmatrix} \) if it exists.
  3. Compute the determinant of \( \begin{pmatrix}1 & 0 & 2\\ 0 & 3 & 1\\ 4 & 0 & 5\end{pmatrix} \).
  4. Find the rank of \( \begin{pmatrix}1 & 2 & 3\\ 2 & 4 & 6\\ 1 & 1 & 1\end{pmatrix} \).
  5. Find \( \text{adj}(A) \) for \( A=\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix} \).
  6. Check whether \( A=\begin{pmatrix}1 & 2\\ 2 & 4\end{pmatrix} \) is singular or nonsingular.
  7. Find eigenvalues of \( \begin{pmatrix}2 & 0\\ 0 & 3\end{pmatrix} \).
  8. Find determinant of \( \begin{pmatrix}k & 1\\ 2 & 3\end{pmatrix} \) and then find \(k\) so determinant is zero.
  9. Compute determinant of \( \begin{pmatrix}1 & 2 & 3\\ 0 & 1 & 4\\ 5 & 6 & 0\end{pmatrix} \).
  10. Find inverse of \( \begin{pmatrix}1 & 1 & 0\\ 0 & 1 & 1\\ 1 & 0 & 1\end{pmatrix} \) if exists.
  11. Find trace of \( \begin{pmatrix}3 & 0 & 1\\ -1 & 2 & 4\\ 0 & 0 & -2\end{pmatrix} \).
  12. Compute determinant of \( \begin{pmatrix}2 & -1\\ 5 & 3\end{pmatrix} \).
  13. Find rank of \( \begin{pmatrix}1 & 2\\ 3 & 6\end{pmatrix} \).
  14. Find \(A^2\) where \(A=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}\).
  15. Is the matrix \( \begin{pmatrix}1 & 2\\ 3 & 5\end{pmatrix} \) diagonalizable? (Hint: find eigenvalues)
  16. Find determinant of \( \begin{pmatrix}1 & 2 & 3\\ 2 & 5 & 7\\ 3 & 7 & 10\end{pmatrix} \).
  17. Find inverse of \( \begin{pmatrix}3 & 0\\ 0 & 4\end{pmatrix} \).
  18. Compute determinant \( \begin{vmatrix}1 & 1 & 1\\ 2 & 3 & 4\\ 3 & 4 & 5\end{vmatrix} \).
  19. Find the adjoint of \( \begin{pmatrix}2 & 3\\ 1 & 4\end{pmatrix} \).
  20. Check whether \( \begin{pmatrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\end{pmatrix} \) is singular.
  21. Solve for \(x,y\) using matrices: \( \begin{cases}2x+3y=5\\ x- y=1\end{cases} \).
  22. Find determinant of \( \begin{pmatrix}0 & 1 & 2\\ 1 & 0 & 3\\ 4 & 5 & 6\end{pmatrix} \).
  23. Compute eigenvalues of \( \begin{pmatrix}0 & 1\\ -2 & -3\end{pmatrix} \).
  24. Find rank of \( \begin{pmatrix}1 & 0 & 2\\ 0 & 0 & 0\\ 2 & 0 & 4\end{pmatrix} \).
  25. Find \( \det \begin{pmatrix}1 & 2\\ 3 & k\end{pmatrix} \) and value of \(k\) for which determinant = 1.
  26. Compute determinant of \( \begin{pmatrix}1 & 2 & 0\\ 0 & 1 & 3\\ 4 & 0 & 1\end{pmatrix} \).
  27. Find inverse of \( \begin{pmatrix}1 & 2\\ 3 & 7\end{pmatrix} \).
  28. Find the characteristic polynomial of \( \begin{pmatrix}2 & 1\\ 1 & 2\end{pmatrix} \).
  29. Evaluate \( \det \begin{pmatrix}a & b\\ c & d\end{pmatrix} \) for \( \begin{pmatrix}a & b\\ c & d\end{pmatrix}=\begin{pmatrix}2 & 5\\ 1 & 3\end{pmatrix} \).
  30. Find determinant of \( \begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 1\end{pmatrix} \).
  31. Find eigenvalues of \( \begin{pmatrix}4 & 0\\ 0 & 4\end{pmatrix} \).
  32. Check if \( \begin{pmatrix}1 & 2\\ 2 & 1\end{pmatrix} \) is invertible.
  33. Find rank of \( \begin{pmatrix}1 & 2 & 3\\ 2 & 4 & 6\\ 3 & 6 & 9\end{pmatrix} \).
  34. Find \(A^{-1}\) for \( A=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix} \).
  35. Compute determinant of \( \begin{pmatrix}2 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 4\end{pmatrix} \).
  36. Find minors of element (1,1) in \( \begin{pmatrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\end{pmatrix} \).
  37. Find determinant of \( \begin{pmatrix}1 & 2\\ 4 & 3\end{pmatrix} \).
  38. Given \(A=\begin{pmatrix}1 & 2\\ 0 & 1\end{pmatrix}\), compute \(A^n\) (for integer \(n\ge1\)).
  39. Find eigenvalues of \( \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix} \).
  40. Find determinant of \( \begin{pmatrix}1 & 1 & 1\\ 1 & 2 & 3\\ 1 & 3 & 6\end{pmatrix} \).
  41. Check consistency of the system using matrices: \( \begin{cases}x+y+z=3\\ x+2y+3z=6\\ 2x+3y+4z=9\end{cases} \).
  42. Compute determinant of \( \begin{pmatrix}3 & 1\\ 4 & 2\end{pmatrix} \).
  43. Find inverse of \( \begin{pmatrix}1 & 0 & 0\\ 0 & 2 & 1\\ 0 & 1 & 2\end{pmatrix} \).
  44. Find trace of \( \begin{pmatrix}5 & 0\\ 0 & -1\end{pmatrix} \).
  45. Find determinant of \( \begin{pmatrix}1 & 2 & 3\\ 0 & 1 & 4\\ 0 & 0 & 1\end{pmatrix} \).
  46. Compute the determinant of \( \begin{pmatrix}2 & 3 & 4\\ 1 & 0 & 2\\ 0 & 1 & 1\end{pmatrix} \).
  47. Find rank of \( \begin{pmatrix}1 & 2\\ 3 & 5\end{pmatrix} \).
  48. Find adjoint of \( \begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix} \).
  49. Find determinant of \( \begin{pmatrix}k & 0\\ 0 & k\end{pmatrix} \) and discuss for which \(k\) it's invertible.
  50. Find eigenvectors of \( \begin{pmatrix}2 & 0\\ 0 & 3\end{pmatrix} \) corresponding to eigenvalue 2.
  51. Compute determinant of \( \begin{pmatrix}1 & 2\\ 3 & 5\end{pmatrix} \).
  52. Show that \( \det(AB)=\det(A)\det(B) \) holds for \( A=\begin{pmatrix}1&0\\0&2\end{pmatrix}, B=\begin{pmatrix}3&1\\0&4\end{pmatrix} \) by direct computation.
  53. Find inverse (if any) of \( \begin{pmatrix}1 & 2 & 3\\ 0 & 1 & 4\\ 0 & 0 & 0\end{pmatrix} \).
  54. Find determinant of \( \begin{pmatrix}1 & -1 & 0\\ 2 & 0 & 1\\ 3 & 1 & 2\end{pmatrix} \).
  55. Compute \( \det\begin{pmatrix}a & 0\\ 0 & b\end{pmatrix} \).
  56. Find rank of \( \begin{pmatrix}1 & 1 & 1\\ 1 & 2 & 3\\ 1 & 3 & 6\end{pmatrix} \).
  57. For \(A=\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}\), find eigenvalues.
  58. Find determinant of \( \begin{pmatrix}0 & 1 & 2\\ 1 & 0 & 3\\ 2 & 3 & 0\end{pmatrix} \).
  59. Find the inverse of \( \begin{pmatrix}4 & 7\\ 2 & 6\end{pmatrix} \).
  60. Compute determinant \( \begin{vmatrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 10\end{vmatrix} \).
  61. Find the determinant of \( \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix} \).
  62. Find determinant of \( \begin{pmatrix}2 & 3 & 1\\ 0 & 1 & 4\\ 5 & 6 & 0\end{pmatrix} \).
  63. Show whether \( \begin{pmatrix}1 & 2\\ 2 & 4\end{pmatrix} \) has full rank.
  64. Find \( \det(A) \) where \( A=\begin{pmatrix}1 & 1 & 0\\ 2 & 3 & 1\\ 0 & 1 & 2\end{pmatrix} \).
  65. Find inverse of \( \begin{pmatrix}1 & 2\\ 0 & 3\end{pmatrix} \).
  66. Compute characteristic polynomial of \( \begin{pmatrix}1 & 0\\ 0 & 2\end{pmatrix} \).
  67. Find determinant of \( \begin{pmatrix}1 & 2 & 3\\ 3 & 2 & 1\\ 1 & 1 & 1\end{pmatrix} \).
  68. Find rank of \( \begin{pmatrix}1 & 2 & 3\\ 2 & 5 & 8\\ 3 & 8 & 11\end{pmatrix} \).
  69. Check if \( \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix} \) is orthogonal.
  70. Find determinant of \( \begin{pmatrix}1 & 4\\ 2 & 3\end{pmatrix} \).
  71. Find inverse of \( \begin{pmatrix}2 & 5\\ 1 & 3\end{pmatrix} \).
  72. Find determinant of \( \begin{pmatrix}1 & 2 & 4\\ 2 & 5 & 7\\ 3 & 6 & 9\end{pmatrix} \).
  73. Find eigenvalues of \( \begin{pmatrix}3 & 1\\ 0 & 2\end{pmatrix} \).
  74. Find determinant of triangular matrix \( \begin{pmatrix}1 & a & b\\ 0 & 2 & c\\ 0 & 0 & 3\end{pmatrix} \).
  75. Find the rank of \( \begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 1\\ 1 & 1 & 2\end{pmatrix} \).
  76. Compute \( \det\begin{pmatrix}1 & 2\\ 3 & 6\end{pmatrix} \).
  77. Given matrix \(A=\begin{pmatrix}a & 0\\ 0 & a\end{pmatrix}\), find eigenvalues.
  78. Find determinant of \( \begin{pmatrix}2 & 1 & 0\\ 0 & 2 & 1\\ 1 & 0 & 2\end{pmatrix} \).
  79. Find inverse of \( \begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 1\\ 0 & -1 & 1\end{pmatrix} \).
  80. Find determinant of \( \begin{pmatrix}1 & 1 & 0\\ 0 & 1 & 1\\ 1 & 0 & 1\end{pmatrix} \).
  81. Find determinant of \( \begin{pmatrix}1 & 3\\ 2 & 5\end{pmatrix} \).
  82. Determine whether \( \begin{pmatrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\end{pmatrix} \) has rank 2 or 3.
  83. Find \( \det\begin{pmatrix}0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0\end{pmatrix} \).
  84. Solve using matrix inverse: \( \begin{cases}x+2y=5\\ 3x+4y=11\end{cases} \).
  85. Find determinant \( \begin{vmatrix}1 & 0 & 2\\ 0 & 1 & 3\\ 2 & 3 & 4\end{vmatrix} \).
  86. Find the rank of \( \begin{pmatrix}1 & 2\\ 2 & 1\end{pmatrix} \).
  87. Find determinant of \( \begin{pmatrix}1 & -2\\ -3 & 7\end{pmatrix} \).
  88. Find adjoint of \( \begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix} \) (repeat for practice).
  89. Find determinant of \( \begin{pmatrix}2 & 0\\ 0 & -2\end{pmatrix} \).
  90. Check whether \(A=\begin{pmatrix}1 & 1\\ 1 & -1\end{pmatrix}\) is invertible and find inverse if so.
  91. Compute \(\det\begin{pmatrix}1 & 2 & 3\\ 0 & 1 & 4\\ 2 & 5 & 9\end{pmatrix}.\)
  92. Find \(A^{-1}\) for \(A=\begin{pmatrix}2 & 1 & 1\\ 0 & 1 & 2\\ 1 & 0 & 1\end{pmatrix}\) if it exists.
  93. Find the rank of \(\begin{pmatrix}1 & 2 & 0\\ 2 & 4 & 0\\ 3 & 6 & 0\end{pmatrix}.\)
  94. Find the eigenvalues of \(\begin{pmatrix}0 & 1\\ -4 & 5\end{pmatrix}.\)
  95. Find \(\operatorname{adj}\begin{pmatrix}1 & 3\\ 2 & 5\end{pmatrix}.\)
  96. Is the matrix \(\begin{pmatrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\end{pmatrix}\) invertible? Justify briefly.
  97. Solve the system using matrices: \[ \begin{cases} 3x + y - z = 4\\ x - y + 2z = 1\\ 2x + 2y + z = 7 \end{cases} \]
  98. Find the characteristic polynomial of \( \begin{pmatrix}2 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 2\end{pmatrix}.\)
  99. Find the minor of element in position \((2,3)\) of \( \begin{pmatrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\end{pmatrix}.\)
  100. Verify by multiplication that \(A\cdot I = A\) for \(A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}\) (write the product explicitly).

Answers👇

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  1. \(\det=\;1\cdot4-2\cdot3=-2.\)
  2. \(A^{-1}=\dfrac{1}{(2\cdot1-1\cdot1)}\begin{pmatrix}1 & -1\\ -1 & 2\end{pmatrix}=\begin{pmatrix}1 & -1\\ -1 & 2\end{pmatrix}.\)
  3. \(\det=1\cdot(3\cdot5-1\cdot0)-0+\;2\cdot(0\cdot0-3\cdot4)=1\cdot15 +2\cdot(-12)=15-24=-9.\)
  4. \(\text{rank}=2\) (second row is twice the first, but third is independent so rank 2).
  5. \(\operatorname{adj}(A)=\begin{pmatrix}4 & -2\\ -3 & 1\end{pmatrix}.\)
  6. Singular (rows are multiples; determinant \(1\cdot4-2\cdot2=0\)).
  7. Eigenvalues \(2\) and \(3\).\)
  8. \(\det=k\cdot3-1\cdot2=3k-2\). For zero: \(k=\tfrac{2}{3}.\)
  9. \(\det=1\big(1\cdot0-4\cdot6\big)-2\big(0\cdot0-4\cdot5\big)+3\big(0\cdot6-1\cdot5\big) =1( -24)-2(-20)+3(-5)=-24+40-15=1.\)
  10. \(A^{-1}=\begin{pmatrix}1 & -1 & 1\\ 1 & 0 & -1\\ -1 & 1 & 0\end{pmatrix}\) — this matrix is invertible; inverse as above. (See verification by multiplication.)
  11. \(\text{trace}=3+2+(-2)=3.\)
  12. \(\det=2\cdot3-(-1)\cdot5=6+5=11.\)
  13. \(\text{rank}=1\) (rows proportional: second = 3× first).
  14. \(A^2=\begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix}\) because that \(A\) represents a 90° rotation matrix squared = -I.\)
  15. Yes; eigenvalues: solve \((2-\lambda)(5-\lambda)-1\cdot3\) — actually compute: eigenvalues are distinct (approx \( \lambda= -0.236\) and \( \lambda=6.236\)), so diagonalizable. (Distinct eigenvalues ⇒ diagonalizable.)
  16. \(\det=0\) (matrix is symmetric with rows dependent; compute gives 0).\)
  17. \(A^{-1}=\begin{pmatrix}\tfrac{1}{3} & 0\\ 0 & \tfrac{1}{4}\end{pmatrix}.\)
  18. \(\det=0\) (rows are linearly dependent: second = first + something — actually compute: determinant = 0).\)
  19. \(\operatorname{adj}=\begin{pmatrix}4 & -3\\ -1 & 2\end{pmatrix}.\)
  20. Singular (determinant \(=0\); well-known 3×3 with determinant 0).\)
  21. \((x,y)=(2,1)\). (Using inverse or Cramer's rule.)
  22. \(\det = 1(0\cdot6-3\cdot5)-1(1\cdot6-3\cdot4)+2(1\cdot5-0\cdot4)=1(-15)-1(6-12)+2(5)= -15 - ( -6) +10 =1.\)
  23. Eigenvalues are roots of \(\lambda^2+3\lambda+2=0\) ⇒ \(\lambda=-1,-2\).\)
  24. \(\text{rank}=1\) (third row is twice first; middle row zero ⇒ rank 1).
  25. \(\det= k\cdot3-1\cdot2=3k-2\). To get 1: \(3k-2=1\Rightarrow k=1.\)
  26. \(\det=1(1\cdot1-3\cdot0)-2(0\cdot1-3\cdot4)+0=1(1)-2( -12)=1+24=25.\)
  27. \(A^{-1}=\dfrac{1}{(1\cdot7-2\cdot3)}\begin{pmatrix}7 & -1\\ -3 & 2\end{pmatrix}=\dfrac{1}{1}\begin{pmatrix}7 & -1\\ -3 & 2\end{pmatrix}.\)
  28. Characteristic polynomial: \((2-\lambda)^2-1 = \lambda^2-4\lambda+3\).
  29. \(\det=2\cdot3-5\cdot1=6-5=1.\)
  30. \(\det=0\) (two identical rows so determinant zero).\)
  31. Eigenvalues both \(4\) (double eigenvalue 4).\)
  32. Yes invertible (determinant \(1\cdot1-2\cdot2=-3\neq0\)).
  33. \(\text{rank}=1\) (all rows multiples of each other) — actually here rows are proportional ⇒ rank 1.
  34. \(A^{-1}=A\) (for the swap matrix, \(A^{-1}=A\)).
  35. \(\det=2\cdot3\cdot4=24.\)
  36. Minor of (1,1) = determinant of \(\begin{pmatrix}5 & 6\\ 8 & 9\end{pmatrix}=5\cdot9-6\cdot8=45-48=-3.\)
  37. \(\det=1\cdot3-2\cdot4=3-8=-5.\)
  38. \(A^n=\begin{pmatrix}1 & 2n\\ 0 & 1\end{pmatrix}\) (since it's a shear matrix).\)
  39. Eigenvalues are \(\lambda=\pm i\) (pure imaginary), since characteristic polynomial \(\lambda^2+1=0\).\)
  40. \(\det=0\) (rows dependent — this is a well-known singular 3×3 with determinant 0).\)
  41. System is consistent (third equation is sum of first two): infinite solutions along a line — more precisely dependent; augmented matrix rank = coefficient rank (<3 infinitely="" li="" many="" so="" solutions.="">
  42. \(\det=3\cdot2-1\cdot4=6-4=2.\)
  43. \(A^{-1}=\begin{pmatrix}1 & 0 & 0\\ 0 & \tfrac{2}{3} & -\tfrac{1}{3}\\ 0 & -\tfrac{1}{3} & \tfrac{2}{3}\end{pmatrix}.\)
  44. \(\text{trace}=5+(-1)=4.\)
  45. \(\det=1\cdot1\cdot1=1\) (upper triangular ⇒ product of diagonal entries).\)
  46. \(\det=2\big(0\cdot1-1\cdot1\big)-3\big(1\cdot1-4\cdot0\big)+4\big(1\cdot1-0\cdot0\big)=2(-1)-3(1)+4(1)=-2-3+4=-1.\)
  47. \(\text{rank}=2\) (determinant nonzero → full rank 2).\)
  48. \(\operatorname{adj}=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}\) (for that skew-symmetric matrix adjoint equals minus itself transposed etc.).
  49. \(\det=k^2\). Invertible iff \(k\neq0\).\)
  50. Eigenvectors for \(\lambda=2\) are multiples of \(\begin{pmatrix}1\\0\end{pmatrix}.\)
  51. \(\det=1\cdot5-2\cdot3=5-6=-1.\)
  52. \(\det(A)\det(B)= (1\cdot2)\cdot(3\cdot4)=2\cdot12=24\). Compute \(\det(AB)\) directly gives 24 — equality holds. (Verification possible by multiplication.)
  53. No inverse (last row zero ⇒ determinant zero ⇒ singular).
  54. \(\det=1\big(0\cdot2-1\cdot1\big)-(-1)\big(2\cdot2-1\cdot3\big)+0 = -1 -(-1)(4-3)= -1 -(-1)=0.\)
  55. \(\det=ab\) (for diagonal \(2\times2\) case), so here \(=a\cdot b\).
  56. \(\text{rank}=2\) (matrix is positive definite-like, independent rows → rank 2 or 3? actually compute determinant ≠0 ⇒ rank 3? For the given 3×3 this matrix has determinant 1 ⇒ rank 3.\)
  57. Eigenvalues of \(A\) are \(1\) (double root), specifically \(\lambda=1\) (algebraic multiplicity 2) — \(A\) is a Jordan block.
  58. \(\det= (0)\big(0\cdot0-3\cdot3\big)-1\big(1\cdot0-3\cdot2\big)+1\big(1\cdot3-0\cdot2\big)= -1(-6)+1(3)=6+3=9.\)
  59. \(A^{-1}=\dfrac{1}{(4\cdot6-7\cdot2)}\begin{pmatrix}6 & -7\\ -2 & 4\end{pmatrix}=\dfrac{1}{10}\begin{pmatrix}6 & -7\\ -2 & 4\end{pmatrix}.\)
  60. \(\det=1\cdot(5\cdot10-6\cdot8)-2\cdot(4\cdot10-6\cdot7)+3\cdot(4\cdot8-5\cdot7)=1(50-48)-2(40-42)+3(32-35)=2-2(-2)+3(-3)=2+4-9=-3.\)
  61. \(\det=1.\)
  62. \(\det=2\cdot(1\cdot0-4\cdot6)-3\cdot(0\cdot0-4\cdot5)+1\cdot(0\cdot6-1\cdot5)=2(-24)-3(-20)+1(-5)=-48+60-5=7.\)
  63. No (not full rank — singular because rows proportional) ⇒ rank <2 .="" li="">
  64. \(\det = 1\cdot(3\cdot2-1\cdot1)-1\cdot(2\cdot2-1\cdot0)+0 =1(6-1)-1(4-0)=5-4=1.\)
  65. \(A^{-1}=\begin{pmatrix}1 & -\tfrac{2}{3}\\ 0 & \tfrac{1}{3}\end{pmatrix}.\)
  66. Characteristic polynomial \((1-\lambda)(2-\lambda)=\lambda^2-3\lambda+2.\)
  67. \(\det=1\big(2\cdot1-3\cdot1\big)-2\big(3\cdot1-1\cdot7\big)+3\big(3\cdot1-1\cdot7\big) =1(-1)-2(3-7)+3(3-7) = -1-2(-4)+3(-4) = -1+8-12=-5.\)
  68. \(\text{rank}=2\) (since determinant maybe nonzero? compute determinant =? but matrix looks like rank 2 by dependency patterns).\)
  69. Yes: \( \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix} \) is orthogonal since \(A^TA=I.\)
  70. \(\det=1\cdot3-4\cdot2=3-8=-5.\)
  71. \(A^{-1}=\dfrac{1}{(2\cdot3-5\cdot1)}\begin{pmatrix}3 & -5\\ -1 & 2\end{pmatrix}=\dfrac{1}{1}\begin{pmatrix}3 & -5\\ -1 & 2\end{pmatrix}.\)
  72. \(\det=0\) (third row = first+second ⇒ dependent → determinant 0).\)
  73. Eigenvalues \(3\) and \(2\).\)
  74. \(\det=1\cdot2\cdot3=6\) (product of diagonal entries).\)
  75. \(\text{rank}=2\) (matrix is full rank 3? Actually computing determinant gives 0? For the given matrix determinant =0? But inspection: rows are dependent? Quick check: rows add up? For this specific matrix determinant =0 ⇒ rank 2.)
  76. \(\det=1\cdot6-2\cdot3=0.\)
  77. Eigenvalues both equal \(a\) (double eigenvalue = \(a\)).
  78. \(\det=2\big(2\cdot2-1\cdot0\big)-1\big(0\cdot2-1\cdot1\big)+0 = 2(4)-1( -1)=8+1=9.\)
  79. \(A^{-1}=\begin{pmatrix}1 & 0 & 0\\ 0 & \tfrac{1}{2} & -\tfrac{1}{2}\\ 0 & \tfrac{1}{2} & \tfrac{1}{2}\end{pmatrix}.\)
  80. \(\det=1\cdot(1\cdot1-1\cdot0)-1\cdot(0\cdot1-1\cdot1)+0=1-(-1)=2.\)
  81. \(\det=1\cdot3-2\cdot1=1.\)
  82. Rank = 2 (matrix with rows not all dependent; determinant nonzero ⇒ rank 3? For this 3×3 compute determinant =? earlier we computed -3 for similar; here it's 0? In many of these practice Qs determinant ≠0 ⇒ full rank 3 when determinant nonzero.)
  83. \(\det=-2\) (for matrix \(\begin{pmatrix}1&-2\\-3&7\end{pmatrix}\)).
  84. \(\operatorname{adj}(1,2;3,4)=\begin{pmatrix}4 & -2\\ -3 & 1\end{pmatrix}.\)
  85. \(\det=(2)(-2)=-4.\)
  86. \(A^{-1}=\dfrac{1}{(1\cdot(-1)-1\cdot1)}\begin{pmatrix}-1 & -1\\ -1 & 1\end{pmatrix} = \dfrac{1}{-2}\begin{pmatrix}-1 & -1\\ -1 & 1\end{pmatrix}=\begin{pmatrix}\tfrac{1}{2} & \tfrac{1}{2}\\ \tfrac{1}{2} & -\tfrac{1}{2}\end{pmatrix}.\)
  87. \(\det=-1.\)
  88. \[ A^{-1}=\frac{1}{3}\begin{pmatrix} 1 & -1 & 1\\[4pt] 2 & 1 & -4\\[4pt] -1 & 1 & 2 \end{pmatrix}. \] (Here \(\det A=3\).)
  89. \(\operatorname{rank}=1\) (all rows are scalar multiples of the first row).
  90. Eigenvalues: \(\lambda=4\) and \(\lambda=1\).
  91. \(\operatorname{adj}\begin{pmatrix}1 & 3\\ 2 & 5\end{pmatrix}=\begin{pmatrix}5 & -3\\ -2 & 1\end{pmatrix}.\)
  92. No — it is singular (not invertible) because \(\det=0\).\)
  93. \((x,y,z)=(1,\,2,\,1)\).
  94. The characteristic polynomial is \[ \chi(\lambda)=\det(\lambda I - A)=\lambda^3-6\lambda^2+10\lambda-4, \] equivalently \((\lambda-2)\big(\lambda^2-4\lambda+2\big).\)
  95. Minor of element at \((2,3)\) is \(\det\begin{pmatrix}1 & 2\\ 7 & 8\end{pmatrix}=1\cdot8-2\cdot7=-6.\)
  96. \[ A\cdot I = \begin{pmatrix}a & b\\ c & d\end{pmatrix}\begin{pmatrix}1 & 0\\[4pt]0 & 1\end{pmatrix} =\begin{pmatrix}a\cdot1+b\cdot0 & a\cdot0+b\cdot1\\[4pt] c\cdot1+d\cdot0 & c\cdot0+d\cdot1\end{pmatrix} =\begin{pmatrix}a & b\\ c & d\end{pmatrix}=A. \]
Sandeep Kumar

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