Advertisements
Advertisements
Question
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)]`
Solution
A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`
∴ AT = `[(1, 3, -2),(2, 2, -3),(4, 1, 2)]`
∴ A + AT = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)] + [(1, 3, -2),(2, 2, -3),(4, 1, 2)]`
= `[(1 + 1, 2 + 3, 4 - 2),(3 + 2, 2 + 2, 1 - 3),(-2 + 4, -3 + 1, 2 + 2)]`
∴ A + AT = `[(2, 5, 2),(5, 4, -2),(2, -2, 4)]`
∴ (A + AT)T = `[(2, 5, 2),(5, 4, -2),(2, -2, 4)]`
∴ (A + AT)T = A + AT i.e., A + AT = (A + AT)T
∴ A + AT is a symmetric matrix.
A – AT = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)] - [(1, 3, -2),(2, 2, -3),(4, 1, 2)]`
= `[(1 - 1, 2 - 3, 4 + 2),(3 - 2, 2 - 2, 1 + 3),(-2 - 4, -3 - 1, 2 - 2)]`
∴ A – AT = `[(0, -1, 6),(1, 0, 4),(-6, -4, 0)]`
∴ (A – AT)T = `[(0, 1, -6),(-1, 0, -4),(6, 4, 0)]`
= `-[(0, -1, 6),(1, 0, 4),(-6, -4, 0)]`
∴ (A – AT)T = – (A – AT)
i.e., A – AT = – (A – AT)T
∴ A – AT is a skew symmetric matrix.
APPEARS IN
RELATED QUESTIONS
Find the values of x and y if
`2 [(x,5),(7,y-3)] [(3,-4),(1,2)] = [(7,6),(15,14)]`
Solve the following equations by reduction method:
x+ y+z = 6,
3x-y+3z = 10
5x+ y-4z = 3
Simplify the following :
`{3 [(1,2,0),(0,-1,3)] - [(1,5,-2),(-3,-4,4)]} [(1),(2),(1)]`
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 = `[(1, -2),(3, -5),(-6, 0)],"B" = [(-1, -2),(4, 2),(1, 5)] "and C" = [(2, 4),(-1, -4),(-3, 6)]`, find the matrix X such that 3A – 4B + 5X = C.
For each of the following matrices, find its transpose and state whether it is symmetric, skew- symmetric or neither.
`[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]`
Construct the matrix A = [aij]3×3 where aij = i − j. State whether A is symmetric or skew-symmetric.
Find x and y, if `[(2x + y, -1, 1),(3, 4y, 4)] [(-1, 6, 4),(3, 0, 3)] = [(3, 5, 5),(6, 18, 7)]`.
If `[(2"a" + "b", 3"a" - "b"),("c" + 2"d", 2"c" - "d")] = [(2, 3),(4, -1)]`, find a, b, c and d.
There are two book shops own 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)][("Suresh"), ("Ganesh")]`
August Sales (in Rupees :
B = `[(6650, 7055, 8905),(7000, 7500, 10200)][("Suresh"), ("Ganesh")]`
If both book shops get 10% profit in the month of August 2017, find the profit for each book seller in each subject in that month.
If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(2, 1),(4, -1),(-3, 3)], "C" = [(1, 2),(-1, 4),(-2, 3)]`, then show that (A + B)T = AT + BT.
If A = `[(5, 4),(-2, 3)]` and B = `[(-1, 3),(4, -1)]`, then find CT, such that 3A – 2B + C = I, whre I is e unit matrix of order 2.
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.
Express each of the following matrix as the sum of a symmetric and a skew symmetric matrix `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`.
If A = `[(2, -1),(3, -2),(4, 1)] "and B" = [(0, 3, -4),(2, -1, 1)]`, verify that (AB)T = BTAT.
Choose the correct alternative.
The matrix `[(0, 0, 0),(0, 0, 0)]` is _______
Choose the correct alternative.
If A = `[(α, 4),(4, α)]` and |A3| = 729, then α = ______.
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 :
If A is symmetric, then A = –AT.
Find a, b, c if `[(1, 3/5, "a"),("b", -5, -7),(-4, "c", 0)]` is a symmetric matrix.
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.
If = `[(2"a" + "b", 3"a" - "b"),("c" + 2"d", 2"c" - "d")] = [(2, 3),(4, -1)]`, find a, b, c and d.
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,
If both book shops got 10 % profit in the month of August 2017, find the profit for each book seller in each subject in that month
Evaluate: `[(3),(2),(1)][(2,-4,3)]`
Evaluate : `[2 -1 3][(4),(3),(1)]`
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:
If A and B are two square matrices of order 3, then (AB)T = ______
State whether the following statement is True or False:
Every square matrix of order n can be expressed as sum of symmetric and skew symmetric matrix
In a Skew symmetric matrix, all diagonal elements are ______
Find x, y, z if `{5[(0, 1),(1, 0),(1, 1)] - [(2, 1),(3, -2),(1, 3)]}[(2),(1)] = [(x + 1),(y - 1), (3z)]`
If A = `[(2, 5),(1, 3)]` then A–1 = ______.
If A = `[(5, 4),(-2, 3)]` and B = `[(-1, 3),(4, -1)]`, then find CT , such that 3A – 2B + C = I, where I is the unit matrix of order 2
