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
\(|adj(24A)|=|adj(3adj(24A))|\)
\(⇒ |24A|^2=|3adj(2A)|^2\)
\(⇒(24^3)^2⋅|A|^2=(3^3)^2|adj(2A)|^2\)
\(⇒24^6⋅|A|^2=3^6|2A|^4\)
\(⇒24^6|A|^2=3^6⋅(2^3)^4|A|^4\)
\(⇒|A|^2=\frac{24^6}{3^6.2^{12}}=\frac{2^{18}.3^6}{3^6.2^{12}}=2^6\)
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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.