The number of singular matrices of order 2, whose elements are from the set $ \{2, 3, 6, 9\} $ is:
Show Hint
Correct Answer: 36
Approach Solution - 1
Let the general form of a \(2 \times 2\) matrix be: \[ \begin{bmatrix} a & b c & d \end{bmatrix} \] The matrix is singular if its determinant is zero: \[ \det = ad - bc = 0 \Rightarrow ad = bc \] Each entry \( a, b, c, d \) is chosen from the set \( \{2, 3, 6, 9\} \), which has 4 elements.
The total number of \(2 \times 2\) matrices that can be formed is: \[ 4^4 = 256 \] We now count how many of these satisfy \( ad = bc \).
We do this by checking all possible 4-tuples \( (a, b, c, d) \in \{2, 3, 6, 9\}^4 \), and count those for which \( ad = bc \).
Using brute-force checking (e.g., via code or enumeration), we find that: \[ \text{Number of singular matrices} = 36 \]
Approach Solution -2
Given determinant: \[ \begin{vmatrix} a & d \\ b & c \end{vmatrix} = ab - bc = 0 \Rightarrow ad = bc \] Case I: Exactly 1 number is used All matrices will be singular. \[ \Rightarrow {}^4C_1 = 4 \] Case II: Exactly 2 numbers are used \[ {}^4C_2 \times 2 \times 2 = 6 \times 4 = 24 \] However, only those with \(ad = bc\) will be singular. So, 6 matrices possible. Case III: Exactly 3 numbers are used None will be singular. \[ \Rightarrow 0 \text{ matrices.} \] --- ### Case IV: Exactly 4 numbers are used For \(ab = cd\): \[ 2 \times 9 = 3 \times 6 \] \[ \Rightarrow {}^4C_1 \times 2! = 8 \text{ matrices.} \] --- Therefore, \[ 4 + 24 + 0 + 8 = 36 \] \[ \boxed{\text{Total number of singular matrices = 36}} \]
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Determinants Questions
- If \[ f(x) = \begin{vmatrix} 2 \cos^4 x & 2 \sin^4 x & 3 + \sin^2 2x \\ 3 + 2 \cos^4 x & 2 \sin^4 x & \sin^2 2x \\ 2 \cos^4 x & 3 + 2 \sin^4 x & \sin^2 2x \end{vmatrix} \] then \( \frac{1}{5} f'(0) \) is equal to
- If \( \sin\left(\frac{y}{x}\right) = \log_e |x| + \frac{\alpha}{2} \) is the solution of the differential equation \[x \cos\left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x\]and \( y(1) = \frac{\pi}{3} \), then \( \alpha^2 \) is equal to
- If \[ f(x) = \begin{vmatrix} x^3 & 2x^2 + 1 & 1 + 3x \\ 3x^2 + 2 & 2x & x^3 + 6 \\ x^3 - x & 4 & x^2 - 2 \end{vmatrix} \] for all \( x \in \mathbb{R} \), then \( 2f(0) + f'(0) \) is equal to
- Let \( A \) be a \( 3 \times 3 \) matrix and \( \det(A) = 2 \). If \(n = \det(\text{adj}(\text{adj}(\ldots(\text{adj}(A))\ldots)))\) with adjoint applied 2024 times, then the remainder when \( n \) is divided by 9 is equal to \(\_\_\_\_\_.\)
If $ A = \begin{pmatrix} 2 & 2 + p & 2 + p + q \\ 4 & 6 + 2p & 8 + 3p + 2q \\ 6 & 12 + 3p & 20 + 6p + 3q \end{pmatrix} $, then the value of $ \det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n $, then $ m + n $ is equal to:
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.