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?
None
To determine the number of correct statements, we need to inspect each statement regarding the properties of non-singular matrices \(A\) and \(B\).
- Statement I: \((AB)^{-1} = A^{-1}B^{-1}\)
The inverse of a product of matrices is given by \((AB)^{-1} = B^{-1} A^{-1}\), not \(A^{-1}B^{-1}\). Therefore, this statement is incorrect.
- Statement II: \((BA)(AB)^{-1} = I\)
If we multiply both sides of the equation \(AB \cdot (AB)^{-1} = I\)by \(BA\), it does not simplify to \(I\). Rather, \((AB)^{-1} \cdot AB = I\)holds true by definition of the inverse. Therefore, this statement is incorrect.
- Statement III: \((AB)^T = A^T B^T\)
The transpose of a product of matrices is given by \((AB)^T = B^T A^T\), not \(A^T B^T\). Therefore, this statement is also incorrect.
None of the statements are correct. Thus, the correct answer is None.
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 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?
- Let $f(x) = \begin{vmatrix} 3x^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}$ where q is any constant, then what is $\frac{d^2}{dx^2}(f(x))$ at $x = 0$ equal to ?
