Advertisements
Advertisements
Question
If A = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`, find A–1. Using A–1, solve the system of linear equations x – 2y = 10 , 2x – y – z = 8 , –2y + z = 7.
Solution
We have, A = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`
Co-factors are:
A11 = –3
A12 = 2
A13 = 2
A31 = –4
A32 = 2
A33 = 3
∴ adjA = `[(-3, 2, 2),(-2, 1, 1),(-4, 2, 3)]^"T"`
= `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`
|A| = 1(–3) – 2(–2) + 0 = 1
∴ `"A"^-1 ("adj A")/|"A"| = [(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`
Now the system of linear equations is
x – 2 = 10
2x– y – z = 8
And –2y + z = 7
or AX = B
i.e., `[(1, -2, 0),(2, -1, -1),(0, -2, 1)][(x),(y),(z)] = [(10),(8),(7)]`
Where, A = `[(1, -2, 0),(2, -1, -1),(0, -2, 1)]`
X = `[(x),(y),(z)]` and B + `[(10),(8),(7)]`
∴ X = `"A"^-1"B"`
⇒ `[(x),(y),(z)] = [(-3, 2, 2),(-2, 1, 1),(-4, 2, 3)] [(10),(8),(7)]`
= `[(-30 + 16 + 14),(-20 +8 + 7),(-40 + 16 + 21)]`
= `[(0),(-5),(-3)]`
∴ x = 0, y = –5 and = –3
APPEARS IN
RELATED QUESTIONS
Write Minors and Cofactors of the elements of following determinants:
`|(2,-4),(0,3)|`
Write Minors and Cofactors of the elements of following determinants:
`|(a,c),(b,d)|`
Write Minors and Cofactors of the elements of following determinants:
`|(1,0,0),(0,1,0),(0,0,1)|`
Using Cofactors of elements of second row, evaluate `triangle = |(5,3,8),(2,0,1),(1,2, 3)|`
If `triangle = |(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|` and Aij is Cofactors of aij, then value of Δ is given by ______.
if A = `((2,3,10),(4,-6,5),(6,9,-20))`, Find `A^(-1)`. Using `A^(-1)` Solve the system of equation `2/x + 3/y +10/z = 2`; `4/x - 6/y + 5/z = 5`; `6/x + 9/y - 20/z = -4`
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}5 & 20 \\ 0 & - 1\end{bmatrix}\]
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}- 1 & 4 \\ 2 & 3\end{bmatrix}\]
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{bmatrix}\]
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{bmatrix}\]
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}0 & 2 & 6 \\ 1 & 5 & 0 \\ 3 & 7 & 1\end{bmatrix}\]
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}a & h & g \\ h & b & f \\ g & f & c\end{bmatrix}\]
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}2 & - 1 & 0 & 1 \\ - 3 & 0 & 1 & - 2 \\ 1 & 1 & - 1 & 1 \\ 2 & - 1 & 5 & 0\end{bmatrix}\]
If \[A = \begin{vmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{vmatrix}\] and Cij is cofactor of aij in A, then value of |A| is given
If Cij is the cofactor of the element aij of the matrix \[A = \begin{bmatrix}2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7\end{bmatrix}\], then write the value of a32C32.
If \[A = \begin{bmatrix}5 & 6 & - 3 \\ - 4 & 3 & 2 \\ - 4 & - 7 & 3\end{bmatrix}\] , then write the cofactor of the element a21 of its 2nd row.
Given A = `[(2, 2, -4),(-4, 2, -4),(2, -1, 5)]`, B = `[(1, -1, 0),(2, 3, 4),(0, 1, 2)]`, find BA and use this to solve the system of equations y + 2z = 7, x – y = 3, 2x + 3y + 4z = 17.
If A is a matrix of order 3 × 3, then number of minors in determinant of A are ______.
The sum of the products of elements of any row with the co-factors of corresponding elements is equal to ______.
If A `= [(0,1,1),(1,0,1),(1,1,0)] "then" ("A"^2 - 3"I")/2 =` ____________.
`abs(("cos" 15°, "sin" 15°),("sin" 75°, "cos" 75°))`
