Advertisements
Advertisements
Question
Solve the following determinant equation:
Solution
\[\text{ Let }∆ = \begin{vmatrix}3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8\end{vmatrix}\]
\[ = \begin{vmatrix}3x - 2 & 3 & 3 \\ 3x - 2 & 3x - 8 & 3 \\ 3x - 2 & 3 & 3x - 8\end{vmatrix} \left[\text{ Applying }C_1 = C_1 + C_2 + C_3 \right]\]
\[ = \left( 3x - 2 \right)\begin{vmatrix}1 & 3 & 3 \\ 1 & 3x - 8 & 3 \\ 1 & 3 & 3x - 8\end{vmatrix} \]
\[ = \left( 3x - 2 \right)\begin{vmatrix}1 & 3 & 3 \\ 0 & 3x - 11 & 0 \\ 1 & 3 & 3x - 8\end{vmatrix} \left[\text{ Applying }R_2 \text{ to }R_2 - R_1 \right]\]
\[ = \left( 3x - 2 \right)\begin{vmatrix}1 & 3 & 3 \\ 0 & 3x - 11 & 0 \\ 0 & 0 & 3x - 11\end{vmatrix} \left[\text{ Applying }R_3 \text{ to }R_3 - R_1 \right]\]
\[ ∆ = \left( 3x - 2 \right) \left( 3x - 11 \right)^2 = 0\]
\[x = \frac{2}{3}, \frac{11}{3}, \frac{11}{3}\]
APPEARS IN
RELATED QUESTIONS
If `|[x+1,x-1],[x-3,x+2]|=|[4,-1],[1,3]|`, then write the value of x.
Solve system of linear equations, using matrix method.
2x + y + z = 1
x – 2y – z =` 3/2`
3y – 5z = 9
Evaluate the following determinant:
\[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\]
Evaluate
\[\begin{vmatrix}2 & 3 & - 5 \\ 7 & 1 & - 2 \\ - 3 & 4 & 1\end{vmatrix}\] by two methods.
Find the value of x, if
\[\begin{vmatrix}3x & 7 \\ 2 & 4\end{vmatrix} = 10\] , find the value of x.
Evaluate the following determinant:
\[\begin{vmatrix}1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sin^2 23^\circ & \sin^2 67^\circ & \cos180^\circ \\ - \sin^2 67^\circ & - \sin^2 23^\circ & \cos^2 180^\circ \\ \cos180^\circ & \sin^2 23^\circ & \sin^2 67^\circ\end{vmatrix}\]
Prove that
\[\begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix} = \left( ab + bc + ca \right)^3\]
Prove the following identity:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\]
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
Prove that :
Prove that :
9x + 5y = 10
3y − 2x = 8
2x − 3y − 4z = 29
− 2x + 5y − z = − 15
3x − y + 5z = − 11
x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10
If A is a singular matrix, then write the value of |A|.
For what value of x, the following matrix is singular?
Write the value of
Find the value of x from the following : \[\begin{vmatrix}x & 4 \\ 2 & 2x\end{vmatrix} = 0\]
If \[A = \begin{bmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}\]. Write the cofactor of the element a32.
The value of the determinant
Using the factor theorem it is found that a + b, b + c and c + a are three factors of the determinant
The other factor in the value of the determinant is
Let \[f\left( x \right) = \begin{vmatrix}\cos x & x & 1 \\ 2\sin x & x & 2x \\ \sin x & x & x\end{vmatrix}\] \[\lim_{x \to 0} \frac{f\left( x \right)}{x^2}\] is equal to
Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5
Solve the following system of equations by matrix method:
3x + y = 7
5x + 3y = 12
Solve the following system of equations by matrix method:
x + y + z = 3
2x − y + z = − 1
2x + y − 3z = − 9
Solve the following system of equations by matrix method:
8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 5
Show that the following systems of linear equations is consistent and also find their solutions:
6x + 4y = 2
9x + 6y = 3
2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0
The number of solutions of the system of equations
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
is
If \[A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix}\] ,find A–1 and hence solve the system of equations x – 2y = 10, 2x + y + 3z = 8 and –2y + z = 7.
If A = `[(2, 0),(0, 1)]` and B = `[(1),(2)]`, then find the matrix X such that A−1X = B.
Solve the following system of equations by using inversion method
x + y = 1, y + z = `5/3`, z + x = `4/3`
Prove that (A–1)′ = (A′)–1, where A is an invertible matrix.
What is the nature of the given system of equations
`{:(x + 2y = 2),(2x + 3y = 3):}`
