Let A and B be two \(3 × 3\) matrices such that AB = I and |A| =\(\frac{1}{8}\). Then |adj (B adj(2A))| is equal to
- 16
- 32
- 64
- 128
The Correct Option is C
Solution and Explanation
A and B are two matrices of order \(3×3\).
and AB = I, |A|=\(\frac{1}{8}\)
Now, |A||B|=\(1\)
|B|=\(8\)
∴\( |adj(B(adj(2A))| \)
\(= |B(adj(2A))|^2\)
= \(|B|^2|adj(2A)|^2 = 2^6|2A|^{2×2 }\)
=\(2^ 6.2^{ 12}.\frac{1}{2 ^{12}}=64\)
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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.