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 \(|2B(\text{adj}(3A))|\)?
This problem asks us to find the value of the determinant \(|2B(\text{adj}(3A))|\) given the determinants of two \(3 \times 3\) matrices, \(A\) and \(B\). We are given that \(|A| = \frac{1}{2\sqrt{2}}\) and \(|B| = \frac{1}{729}\). The order of the matrices is \(n=3\).
Understanding Determinant Properties
To solve this, we need to use several key properties of determinants:
- The determinant of a product of matrices is the product of their determinants: \(|XY| = |X||Y|\).
- The determinant of a scalar multiple of a matrix: \(|kA| = k^n |A|\), where \(n\) is the order of the matrix.
- The determinant of the adjugate of a matrix: \(|\text{adj}(A)| = |A|^{n-1}\).
Step-by-Step Calculation
Let's break down the calculation of \(|2B(\text{adj}(3A))|\) step by step:
Step 1: Apply the Product Rule
Using the property \(|XY| = |X||Y|\), we can write:
\(|2B(\text{adj}(3A))| = |2B| \times |\text{adj}(3A)|\)Step 2: Calculate \(|2B|\)
Here, \(k=2\) and the order \(n=3\). Using the scalar multiple property \(|kA| = k^n |A|\):
\(|2B| = 2^3 |B|\) \(|2B| = 8 \times |B|\)Substituting the given value \(|B| = \frac{1}{729}\):
\(|2B| = 8 \times \frac{1}{729} = \frac{8}{729}\)Step 3: Calculate \(|\text{adj}(3A)|\)
This requires two sub-steps:
- Find the determinant of \(3A\): Using the scalar multiple property \(|kA| = k^n |A|\) with \(k=3\) and \(n=3\): \(|3A| = 3^3 |A|\) \(|3A| = 27 |A|\)
- Find the determinant of the adjugate: Using the property \(|\text{adj}(M)| = |M|^{n-1}\), where \(M = 3A\) and \(n=3\):
\(|\text{adj}(3A)| = |3A|^{3-1}\)
\(|\text{adj}(3A)| = |3A|^2\)
Now substitute the expression for \(|3A|\) from the previous step:
\(|\text{adj}(3A)| = (27 |A|)^2\) \(|\text{adj}(3A)| = 27^2 \times |A|^2\) \(|\text{adj}(3A)| = 729 \times |A|^2\)Substitute the given value \(|A| = \frac{1}{2\sqrt{2}}\):
\(|A|^2 = \left(\frac{1}{2\sqrt{2}}\right)^2 = \frac{1}{(2\sqrt{2})^2} = \frac{1}{4 \times 2} = \frac{1}{8}\)So, the determinant of the adjugate is:
\(|\text{adj}(3A)| = 729 \times \frac{1}{8} = \frac{729}{8}\)
Step 4: Combine the Results
Now substitute the values calculated in Step 2 and Step 3 back into the equation from Step 1:
\(|2B(\text{adj}(3A))| = |2B| \times |\text{adj}(3A)|\) \(|2B(\text{adj}(3A))| = \frac{8}{729} \times \frac{729}{8}\)Step 5: Final Simplification
The terms cancel out:
\(|2B(\text{adj}(3A))| = 1\)Therefore, the value of \(|2B(\text{adj}(3A))|\) is 1.
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. - 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 :- Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct?
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. - 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?
