Let \(\beta\) be a real number Consider the
\(matrix\ A=\begin{pmatrix}\beta & 0 & 1 \\2 & 1 & -2 \\3 & 1 & -2\end{pmatrix}If A^7-(\beta-1) A^6-\beta A^5\) is a singular matrix, then the value of \(9 \beta\) is _____
Let \(\beta\) be a real number Consider the
\(matrix\ A=\begin{pmatrix}\beta & 0 & 1 \\2 & 1 & -2 \\3 & 1 & -2\end{pmatrix}If A^7-(\beta-1) A^6-\beta A^5\) is a singular matrix, then the value of \(9 \beta\) is _____
Correct Answer: 3
Solution and Explanation
The given equation is:
\(|A|^5 |A^2 - (\beta - 1) A - \beta| = 0\)
From the equation, we know that if \( |A| \neq 0 \), we can simplify the equation to:
\(A^2 - (\beta - 1) A - \beta = 0\)
Now, factor the expression:
\(\Rightarrow |A + 1| |A - \beta| = 0\)
Since \( |A + 1| \neq 0 \) (we assume \( A \neq -1 \)), we get:
\(|A - \beta| = 0\)
This implies:
\(A = \beta\)
Now, solving for \( \beta \), we get:
\(\beta = \frac{1}{3} \Rightarrow 9\beta = 3\)
Thus, the value of \( \beta \) is \( \frac{1}{3} \), and \( 9\beta = 3 \).
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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.