Algebra (Matrices and Determinants) Quiz - ICSE Class 12 Mathematics

Free practice quiz: Algebra (Matrices and Determinants) (ICSE Class 12 Mathematics). 62 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 12 Mathematics practice quiz has 62 multiple-choice questions on Algebra (Matrices and Determinants), and 62 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \ 1 & 3 \end{pmatrix}$, then $A+B$ is:

   1. \begin{pmatrix} 3 & 2 \ 4 & 7 \end{pmatrix}
   2. \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}
   3. \begin{pmatrix} 2 & 0 \ 1 & 3 \end{pmatrix}
   4. \begin{pmatrix} 2 & 4 \ 3 & 1 \end{pmatrix}
   Show answer

   Answer: A. \begin{pmatrix} 3 & 2 \ 4 & 7 \end{pmatrix}

   Matrix addition is performed by adding corresponding elements of the matrices. A+B = \begin{pmatrix} 1+2 & 2+0 \ 3+1 & 4+3 \end{pmatrix} = \begin{pmatrix} 3 & 2 \ 4 & 7 \end{pmatrix}

2. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, then $A^T$ (Transpose of A) is:

   1. \begin{pmatrix} 1 & 3 \ 2 & 4 \end{pmatrix}
   2. \begin{pmatrix} 4 & 3 \ 2 & 1 \end{pmatrix}
   3. \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}
   4. \begin{pmatrix} 2 & 1 \ 4 & 3 \end{pmatrix}
   Show answer

   Answer: A. \begin{pmatrix} 1 & 3 \ 2 & 4 \end{pmatrix}

   The transpose of a matrix is obtained by interchanging its rows and columns. The first row becomes the first column and the second row becomes the second column.

3. The system of linear equations $ax + by = c$ and $dx + ey = f$ can be represented in matrix form as:

   1. $\begin{pmatrix} a & b \ d & e \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} c \ f \end{pmatrix}$
   2. $\begin{pmatrix} a & d \ b & e \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} c \ f \end{pmatrix}$
   3. $\begin{pmatrix} a & c \ d & f \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} b \ e \end{pmatrix}$
   4. $\begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} f \ e \end{pmatrix}$
   Show answer

   Answer: A. $\begin{pmatrix} a & b \ d & e \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} c \ f \end{pmatrix}$

   By performing matrix multiplication on the left side, we get $ax + by = c$ and $dx + ey = f$, which matches the given system of equations.

4. Let $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$. Then $A$ is invertible if and only if:

   1. $ad - bc \neq 0$
   2. $ad - bc = 0$
   3. $a+d \neq 0$
   4. $a+d = 0$
   Show answer

   Answer: A. $ad - bc \neq 0$

   A square matrix is invertible if and only if its determinant is non-zero. The determinant of a 2x2 matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$ is $ad - bc$.

5. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, which of the following is equal to $A^{-1}$?

   1. $\frac{1}{-2}\begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix}$
   2. $\frac{1}{10}\begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix}$
   3. $\frac{1}{-2}\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$
   4. $\frac{1}{10}\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$
   Show answer

   Answer: A. $\frac{1}{-2}\begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix}$

   The determinant of $A$ is $(1 imes 4) - (2 imes 3) = 4 - 6 = -2$. The adjugate of $A$ is $\begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix}$. Therefore, $A^{-1} = \frac{1}{\det(A)} adj(A) = \frac{1}{-2}\begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix}$.

More questions in this quiz

Answers and explanations are in the full quiz.

1. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \ 1 & 3 \end{pmatrix}$, then $A-B$ is:
2. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $k=3$, then $kA$ is:
3. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \ 1 & 3 \end{pmatrix}$, then $AB$ is:
4. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, then $A^2$ is:
5. The determinant of the matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ is:
6. Find the determinant of the matrix $A = \begin{pmatrix} 5 & 2 \ 3 & 4 \end{pmatrix}$:
7. If $A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$ (Identity Matrix), then $A^{-1}$ is:
8. For which value of $x$ is the matrix $A = \begin{pmatrix} 3 & x \ 2 & 4 \end{pmatrix}$ singular?
9. If $A = \begin{pmatrix} 2 & 1 \ 1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$, then $A imes B$ is:
10. The trace of a square matrix is defined as:
11. Find the trace of the matrix $A = \begin{pmatrix} 7 & -2 \ 1 & 5 \end{pmatrix}$:
12. If $A$ is a square matrix, then $(A^T)^T$ is equal to:
13. If $A$ and $B$ are matrices of the same order, then $(A+B)^T$ is equal to:
14. If $A$ and $B$ are matrices such that $AB$ is defined, then $(AB)^T$ is equal to:
15. A matrix $A$ is called symmetric if:
16. A matrix $A$ is called skew-symmetric if:
17. If $A = \begin{pmatrix} 0 & 2 \ -2 & 0 \end{pmatrix}$, then $A$ is:
18. The determinant of a diagonal matrix is:
19. If $A$ is a square matrix of order $n$ and $k$ is a scalar, then det($kA$) is equal to:
20. If $A$ is a square matrix, then det($A^T$) is equal to:
21. If $A$ and $B$ are square matrices of the same order, then det($AB$) is equal to:
22. If $A$ is an invertible matrix, then det($A^{-1}$) is equal to:
23. If $AX = B$, where $A$ is an invertible matrix, then the solution for $X$ is:
24. For a system of linear equations represented by $AX=B$, if det(A) $\neq$ 0, then the system has:
25. Which of the following operations is NOT always defined for any two $n imes n$ matrices $A$ and $B$?
26. If $A$ is a $3 imes 2$ matrix and $B$ is a $2 imes 4$ matrix, then the order of the product $AB$ is:
27. If $A$ is a $2 imes 3$ matrix and $B$ is a $3 imes 2$ matrix, then the order of the product $BA$ is:
28. If $A = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{pmatrix}$ and $B = \begin{pmatrix} 7 & 8 \ 9 & 10 \ 11 & 12 \end{pmatrix}$, find the element in the first row and second column of $AB$.
29. If $A = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{pmatrix}$ and $B = \begin{pmatrix} 7 & 8 \ 9 & 10 \ 11 & 12 \end{pmatrix}$, find the element in the first row and second column of $BA$.
30. If $A = \begin{pmatrix} 2 & -1 \ 1 & 3 \end{pmatrix}$, find the inverse of $A$, denoted as $A^{-1}$.
31. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \ 1 & 3 \end{pmatrix}$, is $AB = BA$?
32. For what value of $k$ does the matrix $A = \begin{pmatrix} k & 1 \ 3 & 2 \end{pmatrix}$ have an inverse?
33. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, then det($2A$) is:
34. If $A = \begin{pmatrix} 3 & 1 \ 1 & 2 \end{pmatrix}$, then det($2A$) is:
35. If $A = \begin{pmatrix} 1 & 2 \ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 0 \ 1 & 5 \end{pmatrix}$, what is the determinant of $AB$?
36. If $A = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 0 \ 1 & 5 \end{pmatrix}$, what is the determinant of $AB$?
37. If $A$ is a skew-symmetric matrix of order 3, then det($A$) is:
38. If $A$ is a symmetric matrix, then which of the following is true?
39. If $A = \begin{pmatrix} 2 & 1 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \ 2 & 3 \end{pmatrix}$, then $A-B$ is:
40. If $A = \begin{pmatrix} 1 & -1 \ 2 & 3 \end{pmatrix}$, then $A^2$ is:
41. If $A = \begin{pmatrix} 1 & 1 \ 2 & 1 \end{pmatrix}$, then $A^2$ is:
42. Consider the matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$. What is the condition for $A$ to be idempotent?
43. If $A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$, what is $A^2$?
44. If $A$ is a $3 imes 3$ diagonal matrix with diagonal elements $d_1, d_2, d_3$, then det($A$) is:
45. If $A$ is a $2 imes 2$ matrix and $A^2 = 0$ (the zero matrix), then $A$ is:
46. If $A = \begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix}$, then $A+A^T$ is:
47. If $A = \begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix}$, then $A-A^T$ is:
48. Let $A$ be any square matrix. Which of the following is always a symmetric matrix?
49. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix}$, then $AB$ is:
50. If $A$ is a $2 imes 2$ matrix and $A = \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix}$, what is $A^3$?
51. If $A = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$, then $A^{100}$ is:
52. If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, then $A \cdot adj(A)$ is equal to:
53. If $A$ is a $3 imes 3$ matrix with $\det(A)=2$, then $\det(3A)$ is:
54. If $A$ and $B$ are square matrices of the same order, such that $AB = BA$, then:
55. What is the order of the matrix product $AB$, if $A$ is a $3 imes 4$ matrix and $B$ is a $4 imes 2$ matrix?
56. If $A = \begin{pmatrix} x & y \ z & w \end{pmatrix}$ is a $2 imes 2$ matrix, then $A^2$ is:
57. Let $A$ be a $n imes n$ matrix. If $\det(A) = 0$, then $A$ is:

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