Advertisements
Advertisements
Question
Find A–1 if A = `[(0, 1, 1),(1, 0, 1),(1, 1, 0)]` and show that A–1 = `("A"^2 - 3"I")/2`.
Advertisements
Solution
We have, A = `[(0, 1, 1),(1, 0, 1),(1, 1, 0)]`
Co-factors are:
A11 = –1,
A12 = 1
A13 = 1
A21 = 1
A22 = –1
A23 = 1
A31 = 1
A31 = 1
A32 = 1
A33 = –1
∴ adj A = `[(-1, 1, 1),(1, -1, 1),(1, 1, -1)]^"T"`
= `[(-1, 1, 1),(1, -1, 1),(1, 1, -1)]`
|A| = 0 – 1(–1) + 1.1 = 2
∴ A–1 = `("adj A")/|"A"|`
= `1/2 [(-1, 1, 1),(1, -1, 1),(1, 1, -1)]`
Now, A2 = `[(0, 1, 1),(1, 0, 1),(1, 1, 0)] * [(0, 1, 1),(1, 0, 1),(1, 1, 0)]`
= `[(2, 1, 1),(1, 2, 1),(1, 1, 2)]`
∴ `("a"^2 - 3"I")/2 = 1/2{[(2, 1, 1),(1, 2, 1),(1, 1, 2)] - [(3, 0, 0),(0, 3, 0),(0, 0, 3)]}`
= `1/2 [(-1, 1, 1),(1, -1, 1),(1, 1, -1)]`
= A–1
Hence proved.
APPEARS IN
RELATED QUESTIONS
Write Minors and Cofactors of the elements of the following determinant:
`|(2,-4),(0,3)|`
Write Minors and Cofactors of the elements of the following determinant:
`|(a,c),(b,d)|`
Write Minors and Cofactors of the elements of the following determinant:
`|(1,0,4),(3,5,-1),(0,1,2)|`
Using Cofactors of elements of third column, evaluate Δ = `|(1,x,yz),(1,y,zx),(1,z,xy)|`.
If Δ = `|(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 the 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`
Using matrices, solve the following system of equations :
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3
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:
\[A = \begin{bmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{bmatrix}\]
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
Write the adjoint of the matrix \[A = \begin{bmatrix}- 3 & 4 \\ 7 & - 2\end{bmatrix} .\]
If `"A" = [(1,1,1),(1,0,2),(3,1,1)]`, find A-1. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12.
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.
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°))`
After deleting the \(i\)-th row and \(j\)-th column from a square matrix of order \(n\), what is the order of the remaining matrix used to calculate \(M_{ij}\)?
What calculation gives the resulting value \(M_{ij}\) after the required row and column have been deleted?
After finding the minor \(M_{ij}\), what is the next step to find the cofactor \(C_{ij}\)?
Which multiplication completes the calculation of a cofactor?
For \[\begin{vmatrix}1&-2\\4&3\end{vmatrix}\], what is \(M_{11}\)?
For \[\begin{vmatrix}1&-2\\4&3\end{vmatrix}\], what is \(M_{12}\)?
For \[\begin{vmatrix}1&-2\\4&3\end{vmatrix}\], what is \(M_{21}\)?
For \[\begin{vmatrix}1&-2\\4&3\end{vmatrix}\], what is \(M_{22}\)?
Which expression gives determinant expansion along row \(i\)?
Which expression gives determinant expansion along column \(j\)?
How does the determinant value change when expansion is performed along a different row or column?
For \(i\ne k\), which mixed row/column property is correct?
