TN Online TestSamacheer Kalvi · 1–12

12th Standard Mathematics — Applications of Matrices and Determinants: Online Practice Test

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Q1
If \(|\operatorname{adj}(\operatorname{adj}A)| = |A|^{9}\), then the order of the square matrix \(A\) is
Q2
If \(A\) is a \(3\times3\) non-singular matrix such that \(AA^{T} = A^{T}A\) and \(B = A^{-1}A^{T}\), then \(BB^{T}\) equals
Q3
If \(A = \begin{pmatrix}3 & 5\\ 1 & 2\end{pmatrix}\), \(B = \operatorname{adj}A\) and \(C = 3A\), then \(\dfrac{|\operatorname{adj}B|}{|C|}\) is
Q4
If \(A\begin{pmatrix}1 & -2\\ 1 & 4\end{pmatrix} = \begin{pmatrix}6 & 0\\ 0 & 6\end{pmatrix}\), then \(A\) is
Q5
If \(A = \begin{pmatrix}7 & 3\\ 4 & 2\end{pmatrix}\), then \(9I_{2} - A\) equals
Q6
If \(A = \begin{pmatrix}2 & 0\\ 1 & 5\end{pmatrix}\) and \(B = \begin{pmatrix}1 & 4\\ 2 & 0\end{pmatrix}\), then \(|\operatorname{adj}(AB)|\) is
Q7
If \(P = \begin{pmatrix}1 & x & 0\\ 1 & 3 & 0\\ 2 & 4 & -2\end{pmatrix}\) is the adjoint of a \(3\times3\) matrix \(A\) and \(|A| = 4\), then \(x\) is
Q8
If \(A = \begin{pmatrix}3 & 1 & -1\\ 2 & -2 & 0\\ 1 & 2 & -1\end{pmatrix}\) and \(A^{-1} = (a_{ij})\), then the value of \(a_{23}\) is
Q9
If \(A, B, C\) are invertible matrices of some order, then which one of the following is not true?
Q10
If \((AB)^{-1} = \begin{pmatrix}12 & -17\\ -19 & 27\end{pmatrix}\) and \(A^{-1} = \begin{pmatrix}1 & -1\\ -2 & 3\end{pmatrix}\), then \(B^{-1}\) is
Q11
If \(A^{T}A^{-1}\) is symmetric, then \(A^{2}\) equals
Q12
If \(A\) is a non-singular matrix such that \(A^{-1} = \begin{pmatrix}5 & 3\\ -2 & -1\end{pmatrix}\), then \((A^{T})^{-1}\) is
Q13
If \(A = \begin{pmatrix}\tfrac35 & \tfrac45\\[2pt] x & \tfrac35\end{pmatrix}\) and \(A^{T} = A^{-1}\), then the value of \(x\) is
Q14
If \(A = \begin{pmatrix}1 & \tan\tfrac{\theta}{2}\\ -\tan\tfrac{\theta}{2} & 1\end{pmatrix}\) and \(AB = I_{2}\), then \(B\) equals
Q15
If \(A = \begin{pmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{pmatrix}\) and \(A(\operatorname{adj}A) = \begin{pmatrix}k & 0\\ 0 & k\end{pmatrix}\), then \(k\) is
Q16
If \(A = \begin{pmatrix}2 & 3\\ 5 & -2\end{pmatrix}\) is such that \(\lambda A^{-1} = A\), then \(\lambda\) is
Q17
If \(\operatorname{adj}A = \begin{pmatrix}2 & 3\\ 4 & -1\end{pmatrix}\) and \(\operatorname{adj}B = \begin{pmatrix}1 & -2\\ -3 & 1\end{pmatrix}\), then \(\operatorname{adj}(AB)\) is
Q18
The rank of the matrix \(\begin{pmatrix}1 & 2 & 3 & 4\\ 2 & 4 & 6 & 8\\ -1 & -2 & -3 & -4\end{pmatrix}\) is
Q19
If \(x^{a}y^{b} = e^{m}\), \(x^{c}y^{d} = e^{n}\) and \(\Delta_{1} = \begin{vmatrix}m & b\\ n & d\end{vmatrix}\), \(\Delta_{2} = \begin{vmatrix}a & m\\ c & n\end{vmatrix}\), \(\Delta_{3} = \begin{vmatrix}a & b\\ c & d\end{vmatrix}\), then \(x\) and \(y\) are respectively
Q20
Which of the following is/are correct? (i) Adjoint of a symmetric matrix is symmetric. (ii) Adjoint of a diagonal matrix is diagonal. (iii) If \(A\) is a square matrix of order \(n\) and \(\lambda\) is a scalar, then \(\operatorname{adj}(\lambda A) = \lambda^{n}\operatorname{adj}A\). (iv) \(A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|\,I\).
Q21
If \(\rho(A) = \rho([A\,|\,B])\), then the system of linear equations \(AX = B\) is
Q22
If \(0 \le \theta \le \pi\) and the system \(x + (\sin\theta)y - (\cos\theta)z = 0\), \((\cos\theta)x - y + z = 0\), \((\sin\theta)x + y - z = 0\) has a non-trivial solution, then \(\theta\) is
Q23
The augmented matrix of a system of linear equations is \(\left[\begin{array}{ccc|c}1 & 2 & 7 & 3\\ 0 & 1 & 4 & 6\\ 0 & 0 & \lambda-7 & \mu+5\end{array}\right]\). The system has infinitely many solutions if
Q24
Let \(A = \begin{pmatrix}2 & -1 & 1\\ -1 & 2 & -1\\ 1 & -1 & 2\end{pmatrix}\) and \(4B = \begin{pmatrix}3 & 1 & -1\\ 1 & 3 & x\\ -1 & 1 & 3\end{pmatrix}\). If \(B = A^{-1}\), then the value of \(x\) is
Q25
If \(A = \begin{pmatrix}3 & -3 & 4\\ 2 & -3 & 4\\ 0 & -1 & 1\end{pmatrix}\), then \(\operatorname{adj}(\operatorname{adj}A)\) is

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About this Applications of Matrices and Determinants test

This free online practice test covers Applications of Matrices and Determinants from the 12th Standard Mathematics (Samacheer Kalvi) syllabus. Choose the number of questions and an optional time limit, then answer and submit — everything is checked in your browser, with the correct answers and a worked explanation shown at the end. For the full solutions to every book-back question, see the solved MCQs page.