Advertisements
Advertisements
प्रश्न
Find the cofactor matrix, of the following matrices: `[(5, 8, 7),(-1, -2, 1),(-2, 1, 1)]`
उत्तर
The co-factor Aij of aij is equal to (– 1)i+j Mij.
Here, a11 = 5
∴ M11 = `|(-2, 1),(1, 1)|` = – 2 – 1 = – 3
and A11 = (– 1)1+1 M11 = (1) (– 3) = – 3
a12 = 8
∴ M12 = `|(-1, 1),(-2, 1)|` = – 1 + 2 = 1
and A12 = (– 1)1+2 M11 = (–1) (1) = – 1
a13 = 7
∴ M13 = `|(-1, -2),(-2, 1)|` = – 1 – 4 = – 5
and A13 = (– 1)1+3 M13 = (1) (– 5) = – 5
a21 = – 1
∴ M21 = `|(8, 7),(1, 1)|` = 8 – 7 = 1
and A21 = (– 1)2+1 M21 = (– 1) (1) = – 1
a22 = – 2
∴ M22 = `|(5, 7),(-2, 1)|` = 5 + 14 = 19
and A22 = (– 1)2+2 M22 = (– 1) (19) = 19
a23 = 1
∴ M23 = `|(5, 8),(-2, 1)|` = 5 + 16 = 21
and A23 = (– 1)2+3 M23 = (– 1) (21) = – 21
a31 = – 2
∴ M31 = `|(8, 7),(-2, 1)|` = 8 + 14 = 22
and A31 = (– 1)3+2 M31 = (1) (22) = 22
a32 = 1
∴ M32 = `|(5, 7),(-1, 1)|` = 5 + 7 = 12
and A32 = (– 1)3+2 M32 = (– 1) (12) = – 12
a33 = 1
∴ M33 = `|(5, 8),(-1, -2)|` = – 10 + 8 = – 2
and A33 = (– 1)3+3 M33 = (1) (– 2) = – 2
∴ The matrix of the co-factors is
[Aij]3x3 = `[("A"_11, "A"_12, "A"_13),("A"_21, "A"_22, "A"_23),("A"_31, "A"_32, "A"_33)]`
= `[(-3, -1, -5),(-1, 19, -21),(22, -12, -2)]`.
APPEARS IN
संबंधित प्रश्न
The cost of 4 pencils, 3 pens and 2 erasers is Rs. 60. The cost of 2 pencils, 4 pens and 6 erasers is Rs. 90 whereas the cost of 6 pencils, 2 pens and 3 erasers is Rs. 70. Find the cost of each item by using matrices.
The sum of three numbers is 6. When second number is subtracted from thrice the sum of first and third number, we get number 10. Four times the sum of third number is subtracted from five times the sum of first and second number, the result is 3. Using above information, find these three numbers by matrix method.
Express the following equations in the matrix form and solve them by method of reduction :
2x- y + z = 1, x + 2y + 3z = 8, 3x + y - 4z =1
The cost of 2 books, 6 notebooks and 3 pens is Rs 40. The cost of 3 books, 4 notebooks and 2 pens is Rs 35, while the cost of 5 books, 7 notebooks and 4 pens is Rs 61. Using this information and matrix method, find the cost of 1 book, 1 notebook and 1 pen separately.
Prove that :
Using elementary row operations, find the inverse of the matrix A = `((3, 3,4),(2,-3,4),(0,-1,1))` and hence solve the following system of equations : 3x - 3y + 4z = 21, 2x -3y + 4z = 20, -y + z = 5.
Find the adjoint of the following matrices : `[(2, -3),(3, 5)]`
Fill in the blank :
Order of matrix `[(2, 1, 1),(5, 1, 8)]` is _______
State whether the following is True or False :
Single element matrix is row as well as column matrix.
Choose the correct alternative:
If A = `[(1, 2),(2, -1)]`, then adj (A) = ______
Matrix `[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]` is a singular
The suitable elementary row transformation which will reduce the matrix `[(1, 0),(2, 1)]` into identity matrix is ______
If A is a 3 × 3 matrix and |A| = 2, then the matrix represented by A (adj A) is equal to.
Find the values of a and b if A = B, where A = `[("a" + 4, 3"b"),(8, -6)]`, B = `[(2"a" + 2, "b"^2 + 2),(8, "b"^2 - 5"b")]`
Find x, y, z if A = `[(0, 2y, z),(x, y, -z),(x, -y, z)]` satisfies A′ = A–1.
If possible, using elementary row transformations, find the inverse of the following matrices
`[(2, -1, 3),(-5, 3, 1),(-3, 2, 3)]`
If A = `1/pi [(sin^-1(xpi), tan^-1(x/pi)),(sin^-1(x/pi), cot^-1(pix))]`, B = `1/pi [(-cos^-1(x/pi), tan^-1 (x/pi)),(sin^-1(x/pi),-tan^-1(pix))]`, then A – B is equal to ______.
If (AB)′ = B′ A′, where A and B are not square matrices, then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.
If f(x) = `|(1 + sin^2x, cos^2x, 4 sin 2x),(sin^2x, 1 + cos^2x, 4 sin 2x),(sin^2 x, cos^2 x, 1 + 4 sin 2x)|`
What is the maximum value of f(x)?
if `A = [(2,5),(1,3)] "then" A^-1` = ______
