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 $|M|$.
Important Questions from Matrices and Determinants
What is the value of the determinant of the inverse of the matrix $\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}$?
In obtaining the solution of the system of equations $x + y + z = 7$, $x + 2y+3z = 16$ and $x + 3y+4z = 22$ by Cramer's rule, the value of $y$ is obtained by dividing $D$ by $D_2$, where
$D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 4 \end{vmatrix}$
What is the value of the determinant $D_2$?
Consider the following in respect of non-singular matrices $A$ and $B$ :
I. $(AB)^{-1} = A^{-1}B^{-1}$
II. $(BA)(AB)^{-1} = I$, where $I$ is the identity matrix
III. $(AB)^T = A^T B^T$
How many of the above are correct?
Consider the following statements :
Statement-I :
If $X$ is an $n \times n$ matrix, then $\det(mX) = m^n \det(X)$, where $m$ is a scalar.
Statement-II :
If $Y$ is a matrix obtained from $X$ by multiplying any row or column by a scalar $m$, then $\det(Y) = m\det(X)$.
Which one of the following is correct in respect of the above statements?
Consider the following statements about the matrix $M = \begin{vmatrix} 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 \end{vmatrix}$
Statement-I : The inverse of $M$ does not exist.
Statement-II : $M$ is non-singular.
Which one of the following is correct in respect of the above statements?
