Let A be a 3 × 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to :
66
212
26
- 1
The Correct Option is A
Solution and Explanation
The correct answer is (C) : 26
\(|adj(24A)|=|adj(3adj(24A))|\)
\(⇒ |24A|^2=|3adj(2A)|^2\)
\(⇒(24^3)^2⋅|A|^2=(3^3)^2|adj(2A)|^2\)
\(⇒24^6⋅|A|^2=3^6|2A|^4\)
\(⇒24^6|A|^2=3^6⋅(2^3)^4|A|^4\)
\(⇒|A|^2=\frac{24^6}{3^6.2^{12}}=\frac{2^{18}.3^6}{3^6.2^{12}}=2^6\)
Top Questions on Matrices
- Find x, y, z if
\[ \begin{bmatrix} 5 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 0 & 1 & -2 \\ 1 & -2 & 3 \\ -1 & 1 & 1 \end{bmatrix} \begin{bmatrix} x-1 \\ y+1 \\ 2z \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}. \] - For all \( n \in \mathbb{N} \), if \( 1^3 + 2^3 + 3^3 + \cdots + n^3>x \), then a value of \( x \) among the following is:
- If $A$ and $B$ are two square matrices each of order 3 with $|A| = 3$ and $|B| = 5$, then $|2AB|$ is:
- If \( \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix} \) is a scalar matrix, then \( x^y \) is equal to:
- If $M$ and $N$ are square matrices of order 3 such that $\det(M) = m$ and $MN = mI$, then $\det(N)$ is equal to :
Questions Asked in JEE Main exam
Nature of compounds TeO₂ and TeH₂ is___________ and ______________respectively.
- JEE Main - 2025
- Inorganic chemistry
- Let \( A = [a_{ij}] \) be a matrix of order 3 \(\times\) 3, with \(a_{ij} = (\sqrt{2})^{i+j}\). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), where \(\alpha, \beta \in \mathbb{Z}\), then \(\alpha + \beta\) is equal to:
- JEE Main - 2025
- Matrices and Determinants
Consider the following sequence of reactions :
Molar mass of the product formed (A) is ______ g mol\(^{-1}\).- JEE Main - 2025
- Organic Chemistry
The magnitude of heat exchanged by a system for the given cyclic process ABC (as shown in the figure) is (in SI units):
- JEE Main - 2025
- Electric charges and fields
- The value of \( (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) \) is:
- JEE Main - 2025
- Trigonometric Identities
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.