Question:
The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is $3$ and the trace of $A^{3}$ is $-18$, then the value of the determinant of $A$ is ______
The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is $3$ and the trace of $A^{3}$ is $-18$, then the value of the determinant of $A$ is ______
Updated On: May 6, 2024
Hide Solution
Verified By Collegedunia
Correct Answer: 5
Solution and Explanation
Determinant A =5
Was this answer helpful?
0
1
Top Questions on Matrices
- Four friends Abhay, Bina, Chhaya, and Devesh were asked to simplify \( 4 AB + 3(AB + BA) - 4 BA \), where \( A \) and \( B \) are both matrices of order \( 2 \times 2 \). It is known that \( A \neq B \) and \( A^{-1} \neq B \). Their answers are given as:
- Given \[ A = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix} \] find \( AB \). Hence, solve the system of linear equations: \[ x - y + z = 4, \] \[ x - 2y - 2z = 9, \] \[ 2x + y + 3z = 1. \]
- If \[ A = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix} \] then find \( A^{-1} \). Hence, solve the system of linear equations: \[ x - 2y = 10, \] \[ 2x - y - z = 8, \] \[ -2y + z = 7. \]
- Assertion (A): \( A = \text{diag} [3, 5, 2] \) is a scalar matrix of order \( 3 \times 3 \).
Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix. - If \( A \) and \( B \) are square matrices of order \( m \) such that \( A^2 - B^2 = (A - B)(A + B) \), then which of the following is always correct?
View More Questions
Questions Asked in JEE Advanced exam
- A positive, singly ionized atom of mass number \( A_M \) is accelerated from rest by the voltage \( 192 \, \text{V} \). Thereafter, it enters a rectangular region of width \( w \) with magnetic field \( \vec{B}_0 = 0.1\hat{k} \, \text{T} \). The ion finally hits a detector at the distance \( x \) below its starting trajectory. Which of the following option(s) is(are) correct? \[ \text{(Given: Mass of neutron/proton = } \frac{5}{3} \times 10^{-27} \, \text{kg, charge of the electron = } 1.6 \times 10^{-19} \, \text{C).} \]
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and is in a uniform vertical magnetic field \( B_0 \). When a current \( I \) is passed through the loop, the loop turns about the line PQ by an angle \( \theta \). The angle \( \theta \) is given by:
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A region in the form of an equilateral triangle (in x-y plane) of height \( L \) has a uniform magnetic field \( \mathbf{B} \) pointing in the \( +z \)-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height \( L \), is placed in the x-y plane with its vertex \( P \) at \( x = 0 \) in the orientation shown in the figure. At \( t = 0 \), the loop starts entering the region of the magnetic field with a uniform velocity \( \mathbf{v} \) along the \( +x \)-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graphs best depicts the variation of the induced emf (\( \mathcal{E} \)) in the loop as a function of the distance (\( x \)) starting from \( x = 0 \)?- JEE Advanced - 2024
- Radioactivity
- A table tennis ball has radius \( \frac{3}{2} \times 10^{-2} \, \text{m} \) and mass \( \frac{22}{7} \times 10^{-3} \, \text{kg} \). It is slowly pushed down into a swimming pool to a depth \( d = 0.7 \, \text{m} \) below the water surface and then released from rest. It emerges from the water surface at speed \( v \), without getting wet, and rises to a height \( H \). Which of the following option(s) is(are) correct?\(\text{(Given: } \pi = \frac{22}{7}, g = 10 \, \text{m/s}^2, \text{density of water} = 1 \times 10^3 \, \text{kg/m}^3, \text{viscosity of water} = 1 \times 10^{-3} \, \text{Pa.s).}\)
- JEE Advanced - 2024
- Radioactivity
- An infinitely long thin wire, having a uniform charge density per unit length of \(5 \, \text{nC/m}\), is passing through a spherical shell of radius \(1 \, \text{m}\), as shown in the figure. A \(10 \, \text{nC}\) charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points \(P\) and \(R\), in Volt, is ______. \[ \text{(Given: In SI units } \frac{1}{4\pi\epsilon_0} = 9 \times 10^9, \ln 2 = 0.7). \] Ignore the area pierced by the wire.
- JEE Advanced - 2024
- Radioactivity
View More Questions
Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.