100 Super Revision Test – CBSE Class 12 Determinants (2026)

Note: Important/high-weight questions are highlighted. Try solving before checking final answers below.

1. Define determinant of a square matrix.
2. Find \(\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} \).
3. Evaluate \( \begin{vmatrix} 2 & 5 \\ 1 & 3 \end{vmatrix} \).
4. If two rows of a determinant are interchanged, what happens to its sign?
5. If two rows are identical, what is the value of the determinant?
6. Evaluate \( \begin{vmatrix} 2 & 0 \\ 0 & 2 \end{vmatrix} \).
7. Prove that \( \begin{vmatrix} a & b \\ b & a \end{vmatrix} = a^2 - b^2 \).
8. Find determinant of \( I_2 \).
9. Find \( |3I_2| \).
10. Find \( |0I_2| \).
11. Evaluate
\( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} \).
12. Find determinant of a diagonal matrix with diagonal entries 2, 3, 4.
13. If \(|A| = 5\), find \(|2A|\).
14. What happens to \(|A|\) if a row is multiplied by constant \(k\)?
15. Show that determinant of upper triangular matrix equals product of its diagonal elements.
16. Prove that \( \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} = (b - a)(c - a)(c - b) \).
17. Find \( \begin{vmatrix} x & 1 \\ 1 & x \end{vmatrix} \).
18. Find \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \).
19. For \( \begin{vmatrix} 2 & 3 \\ 4 & 6 \end{vmatrix} \), comment on its value.
20. Find determinant of \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix} \).
21. Find determinant of \( \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} \).
22. Find \( |A| \\) if \\( A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 3 \\ 0 & 0 & 2 \end{bmatrix} \).
23. If \( |A| = 4 \), find \( |A^{-1}| \).
24. Expand \( \begin{vmatrix} 2 & 1 & 0 \\ 0 & 3 & 2 \\ 1 & 0 & 1 \end{vmatrix} \) along first column.
25. Find determinant of \( \begin{vmatrix} 1 & 2 & 1 \\ 0 & 1 & 2 \\ 1 & 1 & 1 \end{vmatrix} \).
26. Find determinant of a zero matrix of order 3.
27. Find determinant of \( A' \\) if \\( |A| = 5 \).
28. Show that \( |A'| = |A| \).
29. If \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), find \( |A'| \).
30. Find \( |AB| \) if \( |A|=2, |B|=3 \).
31. Find \( |A^{-1}| \) if \( |A|=2 \).
32. Find the area of triangle with vertices (1,2), (3,5), (6,7) using determinants.
33. Show that points (1,2), (2,4), (3,6) are collinear using determinant.
34. Find condition for collinearity of (x1, y1), (x2, y2), (x3, y3).
35. If one row is multiplied by zero, what is determinant?
36. Find \( \begin{vmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{vmatrix} \).
37. Find \( \begin{vmatrix} 2 & -3 \\ 1 & 4 \end{vmatrix} \).
38. Find determinant of \( \begin{vmatrix} 1 & 0 & -2 \\ 3 & 1 & 0 \\ 0 & 2 & 1 \end{vmatrix} \).
39. Find \( \begin{vmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{vmatrix} \).
40. Prove that rotation matrix determinant = 1.
41. Find \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} \).
42. Find \( \begin{vmatrix} 2 & 0 & 0 \\ 1 & 3 & 0 \\ 4 & 5 & 6 \end{vmatrix} \).
43. If \( |A|=k \\), find \\( |A^T| \).
44. Evaluate \( \begin{vmatrix} 3 & 2 & 0 \\ 0 & 1 & 4 \\ 1 & 0 & 5 \end{vmatrix} \).
45. Find determinant of \( \begin{vmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{vmatrix} \).
46. Prove that \( |kA| = k^n|A| \) for an \(n×n\) matrix.
47. Find \( |2A| \) if \( |A| = 3 \) and order of A = 2.
48. Find \( |2A| \) if \( |A| = 3 \) and order of A = 3.
49. Find \( |0A| \).
50. Prove that determinant is linear in each row.
51. Find \( |A| \) if A = [[1,2,3],[2,3,4],[3,4,5]].
52. Show that if any two rows are proportional, \(|A| = 0\).
53. Find \( |A| \) if \( A = [[a,b,c],[b,c,a],[c,a,b]] \).
54. Find \( |A| \) if \( A = [[1,1,1],[x,y,z],[x^2,y^2,z^2]] \).
55. Find \( |A| \) if \( A = [[1,p,q],[p,1,r],[q,r,1]] \).
56. Find determinant of \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{vmatrix} \).
57. Find \( |A| \) for A = [[2,1,1],[1,2,1],[1,1,2]].
58. Prove: \( |A| = (a+b+c)(a^2+b^2+c^2 - ab - bc - ca) \) for A = [[1,1,1],[a,b,c],[a^2,b^2,c^2]].
59. Find determinant of [[a, b+c, c+b], [b, c+a, a+c], [c, a+b, b+a]].
60. Find determinant of [[1,0,-1],[0,1,0],[1,1,1]].
61. Find \( |A| \) if \( A = [[1,0,1],[0,1,1],[1,1,0]] \).
62. Find determinant when two rows are equal.
63. Prove that \( |A| = 0 \) if one row is sum of other two.
64. Find determinant of \( [[2,3,4],[0,1,5],[0,0,6]] \).
65. Find determinant of \( [[1,2,3],[2,4,6],[3,6,9]] \).
66. Prove that \( |A| = 0 \) if rows are proportional.
67. Find \( |A| \) when \( A = [[1,2,3],[4,5,6],[7,8,10]] \).
68. Find \( |A| \) for \( A = [[x,1,1],[1,x,1],[1,1,x]] \).
69. Find determinant of \( [[2,1,0],[1,2,1],[0,1,2]] \).
70. Find determinant of [[1,2,3],[1,3,4],[1,4,5]].
71. Find determinant of \( [[x,0,0],[0,x,0],[0,0,x]] \).
72. If \( |A| = 2 \), find \( |3A^{-1}| \).
73. Find determinant of [[2,0,1],[3,4,0],[1,2,3]].
74. Find determinant of [[1,3,1],[0,2,4],[2,1,3]].
75. Find determinant of [[1,0,0],[2,3,0],[1,4,5]].
76. Find determinant of [[0,1,2],[1,2,3],[4,5,6]].
77. Find determinant of [[1,2,1],[3,2,1],[1,1,1]].
78. Find determinant of [[x,1,0],[1,x,1],[0,1,x]].
79. Find determinant of [[1,1,1],[1,2,3],[1,3,6]].
80. Find \( |A| \) when \( A = [[a,b,c],[b,c,a],[c,a,b]] \).
81. Find \( |A| \) for A = [[1,p,q],[p,1,r],[q,r,1]].
82. Find determinant of [[a,b,1],[b,1,a],[1,a,b]].
83. Find \( |A| \) for [[1,2,3],[0,1,4],[5,6,0]].
84. Find \( |A| \) for [[a,b,0],[0,a,b],[b,0,a]].
85. If \( A = [[1,1,1],[1,2,3],[1,4,9]] \), find \( |A| \).
86. Find determinant of [[x,1,1],[1,x,1],[1,1,x]].
87. Find determinant of [[1,a,a^2],[1,b,b^2],[1,c,c^2]].
88. Find determinant of [[1,2,3],[2,4,5],[3,5,7]].
89. Find determinant of [[2,1,0],[0,3,1],[0,0,4]].
90. Find determinant of [[1,2,0],[0,1,3],[4,0,1]].
91. Find determinant of [[1,0,2],[3,1,4],[0,2,5]].
92. Find determinant of [[2,1,3],[0,4,5],[1,0,6]].
93. Find determinant of [[3,0,1],[1,2,0],[4,1,5]].
94. If \(|A| = 3\) and \(|B| = 2\), find \(|A^2B'|\).
95. If \(A\) is singular, what is \(|A|\)?
96. If \(|A| = 0\), what type of matrix is A?
97. If \(|A| = 1\), what type of transformation does it represent?
98. Find \(|\nabla f|\) using determinant for function f(x,y,z).