Advertisements
Advertisements
Question
Using matrices, solve the following system of linear equations :
x + 2y − 3z = −4
2x + 3y + 2z = 2
3x − 3y − 4z = 11
Advertisements
Solution
The system of equations can be written in the form AX = B, where
A `= [(1,2,-3),(2,3,2),(3,-3,-4)],` X`=[("x"),("y"),("z")]` and B =`[(-4),(2),(11)]`
|A| = 1 (-12+6) - 2 (-8 - 6) - 3 (-6 - 9) = 67 ≠ 0
Therefore, A is non singular and so its inverse exists.
A11 = -6, A12 = 14, A13 = -15
A21 = 17, A22 = 5, A23 = 9
A31 = 13, A32 = -8, A33 = -1
Therefore, `"A"^-1 = 1/67[(-6,17,13),(14,5,-8),(-15,9,-1)]`
So X = A-1 B `=1/67[(-6,17,13),(14,5,-8),(-15,9,-1)][(-4),(2),(11)]`
i.e. `[("x"),("y"),("z")]=1/67[(201),(-134),(67)]=[(3),(-2),(1)]`
Hence, x = 3, y = -2 and z = 1
APPEARS IN
RELATED QUESTIONS
Find the value of x, if `|(2,3),(4,5)|=|(x,3),(2x,5)|`.
Without expanding at any stage, find the value of:
`|(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|`
On expanding by first row, the value of the determinant of 3 × 3 square matrix
\[A = \left[ a_{ij} \right]\text{ is }a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}\] , where [Cij] is the cofactor of aij in A. Write the expression for its value on expanding by second column.
Let A = [aij] be a square matrix of order 3 × 3 and Cij denote cofactor of aij in A. If |A| = 5, write the value of a31 C31 + a32 C32 a33 C33.
If A is a 3 × 3 matrix, \[\left| A \right| \neq 0\text{ and }\left| 3A \right| = k\left| A \right|\] then write the value of k.
Which of the following is not correct?
If A is a matrix of order 3 and |A| = 8, then |adj A| = __________ .
Show that Δ = `|(x, "p", "q"),("p", x, "q"),("q", "q", x)| = (x - "p")(x^2 + "p"x - 2"q"^2)`
If Δ = `|(0, "b" - "a", "c" - "a"),("a" - "b", 0, "c" - "b"),("a" - "c", "b" - "c", 0)|`, then show that ∆ is equal to zero.
The value of the determinant ∆ = `|(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)|` = ______.
The determinant ∆ = `|(cos(x + y), -sin(x + y), cos2y),(sinx, cosx, siny),(-cosx, sinx, cosy)|` is independent of x only.
Prove tha `|("bc" - "a"^2, "ca" - "b"^2, "ab" - "c"^2),("ca" - "b"^2, "ab" - "c"^2, "bc" - "a"^2),("ab" - "c"^2, "bc" - "a"^2, "ca" - "b"^2)|` is divisible by a + b + c and find the quotient.
If x + y + z = 0, prove that `|(x"a", y"b", z"c"),(y"c", z"a", x"b"),(z"b", x"c", y"a")| = xyz|("a", "b", "c"),("c", "a", "b"),("b", "c", "a")|`
Let f(t) = `|(cos"t","t", 1),(2sin"t", "t", 2"t"),(sin"t", "t", "t")|`, then `lim_("t" - 0) ("f"("t"))/"t"^2` is equal to ______.
If A = `[(2, lambda, -3),(0, 2, 5),(1, 1, 3)]`, then A–1 exists if ______.
There are two values of a which makes determinant, ∆ = `|(1, -2, 5),(2, "a", -1),(0, 4, 2"a")|` = 86, then sum of these number is ______.
If A is invertible matrix of order 3 × 3, then |A–1| ______.
If A is a matrix of order 3 × 3, then (A2)–1 = ______.
If A and B are matrices of order 3 and |A| = 5, |B| = 3, then |3AB| = 27 × 5 × 3 = 405.
`"A" = abs ((1/"a", "a"^2, "bc"),(1/"b", "b"^2, "ac"),(1/"c", "c"^2, "ab"))` is equal to ____________.
If `"abc" ne 0 "and" abs ((1 + "a", 1, 1),(1, 1 + "b", 1),(1,1,1 + "c")) = 0, "then" 1/"a" + 1/"b" + 1/"c" =` ____________.
Let A be a square matrix of order 2 x 2, then `abs("KA")` is equal to ____________.
For positive numbers x, y, z the numerical value of the determinant `|(1, log_x y, log_x z),(log_y x, 3, log_y z),(log_z x, log_z y, 5)|` is
