If the order of matrix A is \(3 \times 3\) and \(|A|=2\), then the value of \(|3adj (|3A|A2)|\) is?
- \(3^{10}.2^{21}\)
- \(2^{10}.3^{21}\)
- \(2^{12}.3^{15}\)
- \(3^{12}.2^{15}\)
The Correct Option is B
Approach Solution - 1
Given:
-
Step 1: Calculate \( |3A| \):
\( |A| = 2 \)
\( |3A| = 3^3 \cdot |A| \)
\( |3A| = 3^3 \cdot 2 = 27 \cdot 2 \)
-
Step 2: Adj of \( |3A|A^2 \):
\( \text{Adj.}(|3A|A^2) = \text{Adj.}\{(3^3 \cdot 2)A^2\} \)
\( = (2 \cdot 3^3)^2 \cdot (\text{Adj.}A)^2 \)
\( = 2^2 \cdot 36 \cdot (\text{Adj.}A)^2 \)
-
Step 3: Calculate \( |3 \cdot \text{Adj.}(|3A|A^2)| \):
\( |3 \cdot \text{Adj.}(|3A|A^2)| = |2^2 \cdot 36 \cdot (\text{Adj.}A)^2| \)
\( = (2^2 \cdot 3^7)^3 \cdot |\text{Adj.}A|^2 \)
\( = 2^6 \cdot 3^{21} \cdot (|A|^2)^2 \)
\( = 2^6 \cdot 3^{21} \cdot (2^2)^2 \)
\( = 2^{10} \cdot 3^{21} \)
-
Step 4: Final Value:
\( |3 \cdot \text{Adj.}(|3A|A^2)| = 2^{10} \cdot 3^{21} \)
Approach Solution -2
The correct option is (B): \(2^{10}.3^{21}\)
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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.