Cube Roots Unity Determinant Evaluation
This solution details the calculation of the determinant involving p, q, and r, where these represent the cube roots of unity.
Cube Roots of Unity Properties
The cube roots of unity are solutions to the equation \(x^3=1\). These are \(1, \omega, \omega^2\), satisfying:
- \(\omega^3 = 1\)
- \(1 + \omega + \omega^2 = 0\)
Step 1: Assign Values and Compute Squares
To evaluate the determinant, let's assign values to p, q, and r. We choose:
\(p = 1\), \(q = \omega\), \(r = \omega^2\)
Calculate the squares:
- \(p^2 = 1^2 = 1\)
- \(q^2 = \omega^2\)
- \(r^2 = (\omega^2)^2 = \omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega\)
Step 2: Substitute into Determinant
The given determinant is:
\( \Delta = \begin{vmatrix} p^2+q^2 & r^2 & r^2 \\ p^2 & q^2+r^2 & p^2 \\ q^2 & q^2 & r^2+p^2 \end{vmatrix} \)Substitute the computed squares:
\( \Delta = \begin{vmatrix} 1+\omega^2 & \omega & \omega \\ 1 & \omega^2+\omega & 1 \\ \omega^2 & \omega^2 & \omega+1 \end{vmatrix} \)Step 3: Simplify Using Properties
Apply the property \(1 + \omega + \omega^2 = 0\) to simplify terms:
- \(1+\omega^2 = -\omega\)
- \(\omega^2+\omega = -1\)
- \(\omega+1 = -\omega^2\)
The determinant becomes:
\( \Delta = \begin{vmatrix} -\omega & \omega & \omega \\ 1 & -1 & 1 \\ \omega^2 & \omega^2 & -\omega^2 \end{vmatrix} \)Step 4: Evaluate Determinant
Factor \(\omega\) from the first row (R1) and \(\omega^2\) from the third row (R3):
\( \Delta = (\omega) (\omega^2) \begin{vmatrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{vmatrix} \)Using \(\omega^3 = 1\):
\( \Delta = 1 \cdot \begin{vmatrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{vmatrix} \)Calculate the determinant value:
\(\Delta = 1 \times [ -1((-1)(-1) - (1)(1)) - 1((1)(-1) - (1)(1)) + 1((1)(1) - (-1)(1)) ]\)
\(\Delta = [ -1(1 - 1) - 1(-1 - 1) + 1(1 + 1) ]\)
\(\Delta = [ -1(0) - 1(-2) + 1(2) ]\)
\(\Delta = 0 + 2 + 2 = 4\)
Conclusion
The value of the determinant is 4.
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?
