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 __.
\(A=\begin{pmatrix}\beta & 0 & 1 \\2 & 1 & -2 \\3 & 1 & -2\end{pmatrix}\)
If \(A^7-(\beta-1) A^6-\beta A^5\) is a singular matrix, then the value of \(9 \beta\) is __.
Solution and Explanation
Given :
\(A=\begin{bmatrix} \beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2 \end{bmatrix}\)
det(A)=−1 ……(i)
So, For A7 − (β − 1)A6 − βA5 to be singular
|A5| |(A2 − (β − 1)A − β| = 0
⇒ |A5| |(A + I) (A − βI)| = 0 …..(ii)
∴|A5| |A + I| |A − βI| = 0
As we know, |A| ≠ 0
|A+I| or |A−βI| = 0
\(⇒\begin{bmatrix} \beta+1 & 0 & 1 \\ 2 & 2 & -2 \\ 3 & 1 & -1 \end{bmatrix}=0\) {|A + I| ≠ 0}
It is Given that , −1=0 (Rejected)
\(∴ | A − β I | =\begin{vmatrix} 0 & 0 & 1 \\ 2 & 1-\beta & -2 \\ 3 & 1 & -2-\beta \end{vmatrix}=0\)
⇒ 2 − 3(1 − β) = 0
⇒ \(\beta=\frac{1}{3}\)
Therefore, 9β = 3.
So, the correct answer is 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.