The parameter on which the value of the determinant $\Rightarrow \Delta=\begin{vmatrix}1& a & a^2\\ \cos(p-d)x& \cos\ px & \cos(p+d)x\\ \sin(p-d)x& \sin\ px& \sin(p+d)x\\
\end{vmatrix}$ does not depend upon, is
- a
- p
- d
- x
The Correct Option is B
Solution and Explanation
Applying $C_1 \rightarrow C x + C_3$
$=\begin{vmatrix}1+a^2 & a & a^2\\ \cos (p-d)x+\cos(p+d)x& \cos\ px & \cos(p+d)x\\ \sin (p-d)+ \sin(p+d)x& \sin\ px& \sin(p+d)x\\\end{vmatrix}$
$\Rightarrow \Delta=\begin{vmatrix}1+a^2 & a & a^2\\2 \cos\ px\ \cos\ dx& \cos\ px & \cos(p+d)x\\2 \sin\ px\ \sin\ dx& \sin\ px& \sin(p+d)x\\\end{vmatrix}$
Applying $C_1 \rightarrow C_1-2 \cos dx C_2$
$\Rightarrow \Delta=\begin{vmatrix}1+a^2-2a\ \cos\ dx & a & a^2\\0& \cos\ px & \cos(p+d)x\\0& \sin\ px& \sin(p+d)x\\\end{vmatrix}$
$\Rightarrow \Delta= (1 + a^2 - 2a \cos dx) [\sin (p + d) x \cos px - \sin p x \cos (p + d ) x]$
$\Rightarrow \Delta= (1 + a^2 - 2a \cos dx) \sin d x$
which is independent of p.
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Concepts Used:
Applications of Determinants
What is known as Determinants?
The Determinant of a square Matrix is a value ascertained by the elements of a Matrix. In the 2 × 2 Matrix.
The Determinants are calculated by
Det(a b)
The larger Matrices have more complex formulas.
Determinants have different applications throughout Mathematics. For example, they are used in shoelace formulas for calculating the area which is beneficial as a collinearity condition as three collinear points define a triangle that is equal to 0. The Determinant is also used in multiple variable calculi and in computing the cross product of vectors.
Read More: Determinant Formula
Second Method to find the determinant:
The second way to define a determinant is to express in terms of the columns of the matrix by expressing an n x n matrix in terms of the column vectors.
Consider the column vectors of matrix A as A = [ a1, a2, a3, …an] where any element aj is a vector of size x.
Then the determinant of matrix A is defined such that
Det [ a1 + a2 …. baj+cv … ax ] = b det (A) + c det [ a1+ a2 + … v … ax ]
Det [ a1 + a2 …. aj aj+1… ax ] = – det [ a1+ a2 + … aj+1 aj … ax ]
Det (I) = 1
Where the scalars are denoted by b and c, a vector of size x is denoted by v, and the identity matrix of size x is denoted by I.
We can infer from these equations that the determinant is a linear function of the columns. Further, we observe that the sign of the determinant can be interchanged by interchanging the position of adjacent columns. The identity matrix of the respective unit scalar is mapped by the alternating multi-linear function of the columns. This function is the determinant of the matrix.
Properties of Determinant:
- If In is the identity matrix of the order nxn, then det(I) = 1
- If the matrix MT is the transpose of matrix M, then det (MT) = det (M)
- If matrix M-1 is the inverse of matrix M, then det (M-1)
- If two square matrices M and N have the same size, then det (MN) = det (M) det (N)
- If matrix M has a size axa and C is a constant, then det (CM) = Ca det (M)
- If X, Y, and Z are three positive semidefinite matrices of equal size, then the following holds true along with the corollary det (X+Y) ≥ det(X) + det (Y) for X,Y, Z ≥ 0 det (X+Y+Z) + det C ≥ det (X+Y) + det (Y+Z)
- In a triangular matrix, the determinant is equal to the product of the diagonal elements.
- The determinant of a matrix is zero if all the elements of the matrix are zero.
Read More: Properties of Determinants