Determinant Calculation for Matrix Transpose
The problem requires finding the determinant of the product \(AA^T\), where \(A\) is a square matrix with a known determinant \(|A| = -2\). We need to calculate \(|AA^T|\).
Key Determinant Properties Utilized
Two essential properties of determinants are used here:
- Product Rule: The determinant of a product of two square matrices is the product of their determinants. Mathematically, for square matrices \(X\) and \(Y\), \(|XY| = |X| \cdot |Y|\).
- Transpose Rule: The determinant of the transpose of a square matrix is equal to the determinant of the original matrix. Mathematically, for a square matrix \(X\), \(|X^T| = |X|\).
Calculation Steps
- Apply the Product Rule to \(|AA^T|\): \(|AA^T| = |A| \cdot |A^T|\)
- Apply the Transpose Rule (\(|A^T| = |A|\)) to the expression derived in step 1: \(|AA^T| = |A| \cdot |A| = (|A|)^2\)
- Substitute the given value \(|A| = -2\) into the equation: \(|AA^T| = (-2)^2\)
- Compute the final result: \(|AA^T| = 4\)
Thus, the determinant \(|AA^T|\) 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?
