Let \(A\) be a symmetric matrix such that \(|A|=2\) and \(\begin{bmatrix} 2 & 1 \\[0.3em] 3 & \frac 32 \\[0.3em] \end{bmatrix}𝐴−\begin{bmatrix} 1 & 2 \\[0.3em] α & β \\[0.3em] \end{bmatrix}\). If the sum of the diagonal elements of \(A\) is \(s\), then \(\frac {βs}{α^2}\) is equal to ____ .
Correct Answer: 5
Solution and Explanation
(A)Let \( A = \begin{bmatrix} a & b b & c \end{bmatrix} \). Since \( |A| = 2 \): \[ ac - b^2 = 2. \] (B)From the given equation: \[ \begin{bmatrix} 3 & -2 2 & 1 \end{bmatrix} \begin{bmatrix} a & b b & c \end{bmatrix} = \begin{bmatrix} 1 & 2 2 & 7 \end{bmatrix}. \] Expanding row-wise gives equations: \[ 3a - 2b = 1, \quad 3b - 2c = 2, \quad 2a + b = 2, \quad 2b + c = 7. \] (C)Solve these equations to find: \[ a = \frac{3}{4}, \, b = \frac{5}{4}, \, c = \frac{9}{2}. \] Sum of diagonal elements: \[ s = a + c = \frac{3}{4} + \frac{9}{2} = \frac{21}{4}. \] (D)Given \( \alpha = 3 \) and \( \beta = 15 \), compute: \[ \frac{\beta s}{\alpha^2} = \frac{15 \times \frac{21}{4}}{9} = 5. \]
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Let \[ R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \text{ be a non-zero } 3 \times 3 \text{ matrix, where} \]
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Statement II: \( f(x) f(y) = f(x + y) \).
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If \(\alpha = |B|\) and \(\beta\) is the sum of all the diagonal elements of B, then \(\alpha^3 + \beta^3\) is equal to _____. - If \( A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \), \( C = ABA^\top \) and \( X = A^\top C^2 A \), then \( \det (X) \) is equal to:
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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.