Determining Possible Determinant Values for M3 = M
We are given a square matrix M such that \(M^3 = M\). We need to find the number of possible values for its determinant, denoted as \(|M|\).
Using Determinant Properties
- Start with the given equation: \(M^3 = M\).
- Take the determinant of both sides: \(|M^3| = |M|\).
- Apply the determinant property \(|A^n| = |A|^n\): \(|M|^3 = |M|\).
- Let \(x = |M|\). The equation becomes \(x^3 = x\).
- Rearrange and solve for \(x\): \(x^3 - x = 0\) \(x(x^2 - 1) = 0\) \(x(x - 1)(x + 1) = 0\)
- The possible values for \(x\) are \(x = 0\), \(x = 1\), and \(x = -1\).
Counting the Possible Values
The possible values for the determinant \(|M|\) are 0, 1, and -1. These are three distinct values. These values are achievable for specific matrices (e.g., the zero matrix for \(|M|=0\), the identity matrix for \(|M|=1\), and certain matrices with negative eigenvalues for \(|M|=-1\), provided the matrix dimension is odd).
Therefore, there are 3 possible values for \(|M|\).
Similar Questions
- Let X be a matrix of order \(3 \times 3\), Y be a matrix of order \(2 \times 3\) and Z be a matrix of order \(3 \times 2\). Which of the following statements are correct?
I. (ZY)X is defined and is a square matrix of order 3.
II. Y(XZ) is defined and is a square matrix of order 2.
III. X(YZ) is not defined.
Select the answer using the code given below. Let \(A\) and \(B\) be matrices of order \(3 \times 3\)., If \(|A| = \frac{1}{2\sqrt{2}}\) and \(|B| = \frac{1}{729}\), then what is the value of \(|2B(\text{adj}(3A))|\)?
- Consider the following statements in respect of two non-singular matrices \(A\) and \(B\) of the same order \(n\):
1. \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\)
2. \(\text{adj}(AB) = \text{adj}(BA)\)
3. \((AB)\text{adj}(AB) - |AB|I_n\) is a null matrix of order \(n\)
How many of the above statements are correct? - Consider the following statements in respect of a non-singular matrix \(A\) of order \(n\):
1. \(A(\text{adj}A^T) = A(\text{adj}A)^T\)
2. If \(A^2 = A\), then \(A\) is identity matrix of order \(n\)
3. If \(A^3 = A\), then \(A\) is identity matrix of order \(n\)
Which of the statements given above are correct? - Consider the following statements in respect of a skew-symmetric matrix \(A\) of order \(3\):
1. All diagonal elements are zero.
2. The sum of all the diagonal elements of the matrix is zero.
3. \(A\) is orthogonal matrix.
Which of the statements given above are correct? If $A = \begin{bmatrix} \sin 2\theta & 2 \sin^2 \theta - 1 & 0 \\ \cos 2\theta & 2 \sin \theta \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then which of the following statements is/are correct?
1. $A^{-1} = \text{adj}A$
2. $A$ is skew-symmetric matrix
3. $A^{-1} = A^T$
Select the correct answer using the code given below :If
\(\begin{vmatrix}1 + \omega & 1 + \omega^2 & \omega + \omega^2 \\1 & \omega & \omega^2 \\\frac{1}{\omega} & \frac{1}{\omega^2} & 1\end{vmatrix}\)
\(\omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\)
then what isequal to?
- If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?
- What is \(A(\text{adj } A)\) equal to?
- If \(\begin{vmatrix} a-b & p-q & x-y \\ b-c & q-r & y-z \\ c-a & r-p & z-x \end{vmatrix} = k \begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}\) then what is the value of k ?
Important Questions from Matrices and Determinants
- If each element of a matrix is zero, then the matrix is:
- Let X be a matrix of order \(3 \times 3\), Y be a matrix of order \(2 \times 3\) and Z be a matrix of order \(3 \times 2\). Which of the following statements are correct?
I. (ZY)X is defined and is a square matrix of order 3.
II. Y(XZ) is defined and is a square matrix of order 2.
III. X(YZ) is not defined.
Select the answer using the code given below. Let \(A\) and \(B\) be matrices of order \(3 \times 3\)., If \(|A| = \frac{1}{2\sqrt{2}}\) and \(|B| = \frac{1}{729}\), then what is the value of \(|2B(\text{adj}(3A))|\)?
- Consider the following statements in respect of two non-singular matrices \(A\) and \(B\) of the same order \(n\):
1. \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\)
2. \(\text{adj}(AB) = \text{adj}(BA)\)
3. \((AB)\text{adj}(AB) - |AB|I_n\) is a null matrix of order \(n\)
How many of the above statements are correct? - Consider the following statements in respect of a non-singular matrix \(A\) of order \(n\):
1. \(A(\text{adj}A^T) = A(\text{adj}A)^T\)
2. If \(A^2 = A\), then \(A\) is identity matrix of order \(n\)
3. If \(A^3 = A\), then \(A\) is identity matrix of order \(n\)
Which of the statements given above are correct?
