Advertisements
Advertisements
Question
Prove that :
Advertisements
Solution
\[\text{ Let LHS }= \Delta = \begin{vmatrix} x + 4 & x & x\\x & x + 4 & x\\x & x & x + 4 \end{vmatrix}\]
\[ = \begin{vmatrix} 3x + 4 & 3x + 4 & 3x + 4\\x & x + 4 & x\\x & x & x + 4 \end{vmatrix} \left[\text{ Applying }R_1 \to R_1 + R_2 + R_3 \right] \]
\[ = \left( 3x + 4 \right)\begin{vmatrix} 1 & 1 & 1\\x & x + 4 & x \\x & x & x + 4 \end{vmatrix} \left[\text{ Taking out }\left( 3x + 4 \right)\text{ common from }R_1 \right]\]
\[ = \left( 3x + 4 \right)\begin{vmatrix} 1 & 0 & 0\\x & 4 & 0\\x & 0 & 4 \end{vmatrix} \left[\text{ Applying }C_2 \to C_2 - C_1\text{ and }C_3 \to C_3 - C_1 \right]\]
\[ = \left( 3x + 4 \right) \left( 4^2 \right) \left[\text{ Expanding along }R_1 \right]\]
\[ = 16\left( 3x + 4 \right) \]
\[ = RHS\]
APPEARS IN
RELATED QUESTIONS
Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to ______.
Examine the consistency of the system of equations.
x + 3y = 5
2x + 6y = 8
Solve the system of linear equations using the matrix method.
5x + 2y = 4
7x + 3y = 5
Evaluate the following determinant:
\[\begin{vmatrix}x & - 7 \\ x & 5x + 1\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}\]
\[∆ = \begin{vmatrix}\cos \alpha \cos \beta & \cos \alpha \sin \beta & - \sin \alpha \\ - \sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1/a & a^2 & bc \\ 1/b & b^2 & ac \\ 1/c & c^2 & ab\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}\]
\[\begin{vmatrix}b + c & a & a \\ b & c + a & b \\ c & c & a + b\end{vmatrix} = 4abc\]
Find the area of the triangle with vertice at the point:
(−1, −8), (−2, −3) and (3, 2)
Find the value of \[\lambda\] so that the points (1, −5), (−4, 5) and \[\lambda\] are collinear.
Prove that :
3x + y = 5
− 6x − 2y = 9
x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10
Solve each of the following system of homogeneous linear equations.
x + y − 2z = 0
2x + y − 3z = 0
5x + 4y − 9z = 0
Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0
If I3 denotes identity matrix of order 3 × 3, write the value of its determinant.
Write the value of
The value of the determinant
The value of the determinant
The determinant \[\begin{vmatrix}b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ca & c - a & ab - a^2\end{vmatrix}\]
Solve the following system of equations by matrix method:
3x + 4y − 5 = 0
x − y + 3 = 0
Solve the following system of equations by matrix method:
3x + y = 7
5x + 3y = 12
Solve the following system of equations by matrix method:
5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25
Solve the following system of equations by matrix method:
3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2
Show that the following systems of linear equations is consistent and also find their solutions:
2x + 2y − 2z = 1
4x + 4y − z = 2
6x + 6y + 2z = 3
The number of solutions of the system of equations:
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
x + y = 1
x + z = − 6
x − y − 2z = 3
System of equations x + y = 2, 2x + 2y = 3 has ______
If A = `[(2, 0),(0, 1)]` and B = `[(1),(2)]`, then find the matrix X such that A−1X = B.
`abs ((("b" + "c"^2), "a"^2, "bc"),(("c" + "a"^2), "b"^2, "ca"),(("a" + "b"^2), "c"^2, "ab")) =` ____________.
The existence of unique solution of the system of linear equations x + y + z = a, 5x – y + bz = 10, 2x + 3y – z = 6 depends on
For what value of p, is the system of equations:
p3x + (p + 1)3y = (p + 2)3
px + (p + 1)y = p + 2
x + y = 1
consistent?
The system of linear equations
3x – 2y – kz = 10
2x – 4y – 2z = 6
x + 2y – z = 5m
is inconsistent if ______.
