Advertisements
Advertisements
Question
If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + BT)T = AT + B.
Solution
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
RELATED QUESTIONS
If A = `[(1,2),(3,-1)] , "B" = [(7,1),(2,5)]`
Verify that |AB| = |A|.|B|
Simplify the following :
`{3 [(1,2,0),(0,-1,3)] - [(1,5,-2),(-3,-4,4)]} [(1),(2),(1)]`
Solve the following equations by reduclion method
x+3y+3z= 16 , x+4y+4z=21 , x+3y+4z = 19
If A = `[(1,-1,2),(3,0,-2),(1,0,3)]` ,
verify that A (adj A) = (adj A) A = |A| . I
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.
If A = `[(7, 3, 1),(-2, -4, 1),(5, 9, 1)]`, find (AT)T.
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)]`.
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) Physics Chemistry Mathematics
B = `[(6650, 7055, 8905),(7000, 7500, 10200)][("Suresh"), ("Ganesh")]`
Find the increase in sales in Rupees from July to August 2017.
If A = `[(1, 2),(-1, -2)], "B" = [(2, "a"),(-1, "b")]` and (A + B)2 = A2 + B2, find the values of a and b.
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 (BA)T = ATBT.
Choose the correct alternative.
The matrix `[(0, 0, 0),(0, 0, 0)]` is _______
State whether the following is True or False :
Every scalar matrix is unit matrix.
State whether the following is True or False :
If A is symmetric, then A = –AT.
State whether the following is True or False :
If A and B are square matrices of same order, then (A + B)2 = A2 + 2AB + B2.
Find a, b, c if `[(1, 3/5, "a"),("b", -5, -7),(-4, "c", 0)]` is a symmetric 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.
Find matrices A and B, if `2"A" - "B" = [(6, -6, 0),(-4, 2, 1)] and "A" - 2"B" = [(3, 2, 8),(-2, 1, -7)]`
Simplify, `costheta[(costheta, sintheta),(-sintheta, costheta)] + sintheta[(sintheta, -costheta),(costheta, sintheta)]`
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 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.
Evaluate: `[(3),(2),(1)][(2,-4,3)]`
Choose the correct alternative:
`[(3, 2, 1)][(2),(-2),(-1)]` = ______
Choose the correct alternative:
For any square matrix B, matrix B + BT is ______
Choose the correct alternative:
If A and B are two square matrices of order 3, then (AB)T = ______
If A = `[(2, 5),(1, 3)]` then A–1 = ______.
