Advertisements
Advertisements
प्रश्न
If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + BT)T = AT + B.
उत्तर
A = `[(-1, 2, 1),(-3, 2, -3)] "and B" = [(2, 1),(-3, 2),(-1, 3)]`
∴ AT = `[(-1, -3),(2, 2),(1, -3)] "and B"^"T" = [(2, -3, -1),(1, 2, 3)]`
∴ A + BT = `[(-1, 2, 1),(-3, 2, -3)] + [(2, -3, 1),(1, 2, 3)]`
= `[(-1 + 2, 2 - 3, 1 - 1),(-3 + 1, 2 + 2, -3 + 3)]`
= `[(1, -1, 0),(-2, 4, 0)]`
∴ (A + BT)T = `[(1, -2),(-1, 4),(0, 0)]` ...(i)
Now, AT + B = `[(-1, -3),(2, 2),(1, -3)] + [(2, 1),(-3, 2),(-1, 3)]`
= `[(-1 + 2, -3 + 1),(2 - 3, 2 + 2),(1 - 1, -3 + 3)]`
= `[(1, -2),(-1, 4),(0, 0)]` ...(ii)
From (i) and (ii), we get
(A + BT)T = AT + B.
APPEARS IN
संबंधित प्रश्न
Solve the following equations by reduction method:
x + y + z = 6,
3x - y + 3z = 10
5x + y - 4z = 3
If A = `[(1,2),(3,-1)] , "B" = [(7,1),(2,5)]`
Verify that |AB| = |A|.|B|
Find x , y , z , w if `[("x+y","x-y"),("y+z+w","2w-z")]` = `[(2,-1),(9,5)]`
If A = `[(1,2,3),(2,"a",2),(5,7,3)]` is a singular matrix , find the value of 'a'.
If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2),(0, 3)] "and C" = [(4, 3),(-1, 4),(-2, 1)]`, Show that (A + B) + C = A + (B + C)
If A = `[(5, 1, -4),(3, 2, 0)]`, find (AT)T.
Find x, y, z if `[(0, -5i, x),(y, 0, z),(3/2, - sqrt(2), 0)]` is a skew symmetric matrix.
For each of the following matrices, find its transpose and state whether it is symmetric, skew- symmetric or neither.
`[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`
Construct the matrix A = [aij]3×3 where aij = i − j. State whether A is symmetric or skew-symmetric.
Find matrices A and B, if 2A – B = `[(6, -6, 0),(-4, 2, 1)]` and A – 2B = `[(3, 2, 8),(-2, 1, -7)]`.
Find x and y, if `[(2x + y, -1, 1),(3, 4y, 4)] [(-1, 6, 4),(3, 0, 3)] = [(3, 5, 5),(6, 18, 7)]`.
Find AT, if A = `[(1, 3),(-4, 5)]`
If A = `[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`, prove that AT = A.
If A = `[(7, 3, 0),(0, 4, -2)], "B" = [(0, -2, 3),(2, 1, -4)]`, then find AT + 4BT.
If A = `[(7, 3, 0),(0, 4, -2)], "B" = [(0, -2, 3),(2, 1, -4)]`, then find 5AT – 5BT.
If A = `[(1, 0, 1),(3, 1, 2)], "B" = [(2, 1, -4),(3, 5, -2)] "and" "C" = [(0, 2, 3),(-1, -1, 0)]`, verify that (A + 2B + 3C)T = AT + 2BT + CT.
Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`
Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where A = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)]`
If A = `[(2, -1),(3, -2),(4, 1)] "and B" = [(0, 3, -4),(2, -1, 1)]`, verify that (AB)T = BTAT.
Fill in the blank:
A = `[(3),(1)]` is ........................ matrix.
Fill in the blank :
Matrix B = `[(0, 3, 1),(-3, 0, -4),("p", 4, 0)]` is skew symmetric, then the value of p is _______
State whether the following is True or False :
Every scalar matrix is unit matrix.
State whether the following is True or False :
A = `[(4, 5),(6, 1)]` is no singular matrix.
If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2), (0, 3)] and "C" = [(4, 3),(-1, 4),(-2, 1)]` Show that A + B = B + A
If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2), (0, 3)] and "C" = [(4, 3),(-1, 4),(-2, 1)]` Show that (A + B) + C = A + (B + C)
If A = `[(1, -2),(5, 3)], "B" = [(1, -3),(4, -7)]`, then find the matrix A – 2B + 6I, where I is the unit matrix of order 2.
There are two book shops owned by Suresh and Ganesh. Their sales (in Rupees) for books in three subject – Physics, Chemistry and Mathematics for two months, July and August 2017 are given by two matrices A and B.
July sales (in Rupees), Physics Chemistry Mathematics.
A = `[(5600, 6750, 8500),(6650, 7055, 8905)]"First Row Suresh"/"Second Row Ganesh"`
August sales(in Rupees), Physics Chemistry Mathematics
B = `[(6650, 7055, 8905),(7000, 7500, 10200)]"First Row Suresh"/"Second Row Ganesh"` then,
Find the increase in sales in Rupees from July to August 2017.
Answer the following question:
Find matrices A and B, where 3A – B = `[(-1, 2, 1),(1, 0, 5)]` and A + 5B = `[(0, 0, 1),(-1, 0, 0)]`
Choose the correct alternative:
If A = `[(1, 3/5, x),(y, -5, -7),(-4, -7, 0)]` is a symmetric matrix, then the values of x and y are ______ respectively.
Choose the correct alternative:
`[(3, 2, 1)][(2),(-2),(-1)]` = ______
Choose the correct alternative:
For any square matrix B, matrix B + BT is ______
Find k, if A = `[(3, -2),(4, -2)]` and A2 = kA – 2I, where I is identity matrix of order 2
Find the x, y, z, if `{3[(2,0),(0,2),(2,2)]-4[(1,1),(-1,2),(3,1)]}[(1),(2)]=[(x-3),(y-1),( 2z)]`
