Class 12Mathematics3 and 4 .Determinants and MatricesRank of Matrices , Some special determinants, differentiation and integration of determinants
MediumIf $y = \begin{vmatrix} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{vmatrix}$,prove that $\frac{dy}{dx} = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\ l & m & n \\ a & b & c \end{vmatrix}$.
(N/A) Given $y = \begin{vmatrix} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{vmatrix}$.
Expanding the determinant along the first row,we get:
$y = f(x)(mc - nb) - g(x)(lc - na) + h(x)(lb - ma)$.
Differentiating both sides with respect to $x$:
$\frac{dy}{dx} = \frac{d}{dx}[f(x)(mc - nb)] - \frac{d}{dx}[g(x)(lc - na)] + \frac{d}{dx}[h(x)(lb - ma)]$.
Since $l, m, n, a, b, c$ are constants:
$\frac{dy}{dx} = f'(x)(mc - nb) - g'(x)(lc - na) + h'(x)(lb - ma)$.
This expression is the expansion of the determinant where the first row is differentiated:
$\frac{dy}{dx} = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\ l & m & n \\ a & b & c \end{vmatrix}$.
Hence,the result is proved.
Expanding the determinant along the first row,we get:
$y = f(x)(mc - nb) - g(x)(lc - na) + h(x)(lb - ma)$.
Differentiating both sides with respect to $x$:
$\frac{dy}{dx} = \frac{d}{dx}[f(x)(mc - nb)] - \frac{d}{dx}[g(x)(lc - na)] + \frac{d}{dx}[h(x)(lb - ma)]$.
Since $l, m, n, a, b, c$ are constants:
$\frac{dy}{dx} = f'(x)(mc - nb) - g'(x)(lc - na) + h'(x)(lb - ma)$.
This expression is the expansion of the determinant where the first row is differentiated:
$\frac{dy}{dx} = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\ l & m & n \\ a & b & c \end{vmatrix}$.
Hence,the result is proved.
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View SolutionWhich of the following statements is false?
$1$. If $A$ is a skew-symmetric matrix of order $5 \times 5$,then the rank of $A$ is less than $5$.
$2$. If $P$ is a non-zero column matrix and $Q$ is a non-zero row matrix,then the rank of $PQ$ is $1$.
$3$. The rank of $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{bmatrix}$ is $2$.
$4$. If the lines $a_r x + b_r y + c_r = 0$ $(r = 1, 2, 3)$ are distinct and intersect at a point,then the rank of $\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$ is $3$.
$1$. If $A$ is a skew-symmetric matrix of order $5 \times 5$,then the rank of $A$ is less than $5$.
$2$. If $P$ is a non-zero column matrix and $Q$ is a non-zero row matrix,then the rank of $PQ$ is $1$.
$3$. The rank of $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{bmatrix}$ is $2$.
$4$. If the lines $a_r x + b_r y + c_r = 0$ $(r = 1, 2, 3)$ are distinct and intersect at a point,then the rank of $\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$ is $3$.
The rank of the matrix $\begin{bmatrix} 3 & 5 & -1 & 4 \\ 2 & 1 & 3 & -2 \\ 8 & 11 & 1 & 6 \\ -7 & -14 & 6 & -14 \end{bmatrix}$ is
In a matrix $A$,if all the sub-matrices of order $k$ are singular and there is at least one non-singular sub-matrix of order $r$ $(r < k)$,then the rank $(\rho)$ of the matrix $A$:
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