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.
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?
