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?
Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
To determine the correctness of the given statements, let's break down each statement step-by-step:
Statement-I:
The statement claims that for an \(n \times n\) matrix \(X\), if every element of \(X\) is multiplied by a scalar \(m\), the determinant scales by \(m^n\). Mathematically, this is represented as:
\(\det(mX) = m^n \det(X)\)
This is a known property of determinants. If each element of an \(n \times n\) matrix is multiplied by a scalar, the determinant of the resulting matrix is the original determinant multiplied by the scalar raised to the power of the order of the matrix (here \(n\)).
Thus, Statement-I is correct.
Statement-II:
This statement asserts that if matrix \(Y\) is formed by multiplying a single row or column of matrix \(X\) by a scalar \(m\), then:
\(\det(Y) = m\det(X)\)
This is indeed true: Multiplying any single row or column of a matrix by a scalar affects the determinant of the matrix by that scalar.
Thus, Statement-II is correct.
Explanatory Relation:
Statement-II effectively explains the basis for Statement-I: The determinant transformation in Statement-I is a repeated application of the transformation in Statement-II. Specifically, when the entire matrix is multiplied by the scalar, it's equivalent to scaling each row (or column) individually by that scalar, which leads to the overall determinant being scaled by \(m^n\).
Conclusion:
Based on the above analysis, both statements are individually correct. Moreover, Statement-II explains the mechanism through which Statement-I holds true. Therefore, the correct option is:
Both Statement-I and Statement-II are correct and Statement-II explains Statement-I.
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 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 ?
