Let
\( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix} \)
and \(|2A|^3 = 2^{21}\) where \(\alpha, \beta \in \mathbb{Z}\). Then a value of \(\alpha\) is:
Let
\( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix} \)
and \(|2A|^3 = 2^{21}\) where \(\alpha, \beta \in \mathbb{Z}\). Then a value of \(\alpha\) is:
- 5
- 3
- 9
- 17
The Correct Option is A
Solution and Explanation
Step 1: Calculate the Determinant of A
\(|A| = \alpha^2 - \beta^2\)
Step 2: Use the Condition \(|2A|^3 = 2^{21}\)
We know that:
\(|2A| = 2^3 |A| = 2^{21} \Rightarrow |A| = 2^4 = 16\)
Step 3: Set Up the Equation
\(\alpha^2 - \beta^2 = 16\)
Factor as \((\alpha + \beta)(\alpha - \beta) = 16\).
Step 4: Solve for Possible Values of \(\alpha\)
Possible integer solutions for \((\alpha, \beta)\) that satisfy the equation give \(\alpha = 4\) or \(\alpha = 5\).
Since \(\alpha = 5\) satisfies the condition, we choose \(\alpha = 5\).
So, the correct answer is: 5
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- 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:
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- 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?
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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.