Problem 1 :
If A and B are symmetric matrices of same order, prove that
(i) AB + BA is a symmetric matrix.
(ii) AB - BA is a skew-symmetric matrix
Solution :
(i) Since A and B are symmetric matrices, then
A^{T} = A
B^{T} = B
Let P = AB + BA
P^{T} = (AB + BA)^{T}
P^{T} = (AB)^{T} + (BA)^{T}
P^{T} = B^{T}A^{T} + A^{T}B^{T}
P^{T} = BA + AB
P^{T} = AB + BA
P^{T} = P
So, AB + BA is a symmetric matrix.
(ii) Let Q = AB - BA
Q^{T} = (AB - BA)^{T}
Q^{T} = (AB)^{T} - (BA)^{T}
Q^{T} = B^{T}A^{T} - A^{T}B^{T}
Q^{T} = BA - AB
Q^{T} = -(AB - BA)
Q^{T} = -Q
So, AB - BA is skew symmetric matrix.
Problem 2 :
A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins and almonds.
Pack I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds.
Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds.
Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds.
The cost of 50 gm of cashew nuts is $50, 50 gm of raisins is $10, and 50 gm of almonds is $60. What is the cost of each gift pack?
Solution :
By using the given items, we may construct a matrix. Then construct another matrix with cost per grams.
By multiplying the above matrices, we may get the cost of each pack.
Cost of 50 gm of cashew nuts is $50
Cost of 1 gm of cashew nuts is 50/50 = 1
Cost of 50 gm of raisins is $10
Cost 1 gm of raisins is 10/50 = 1/5
Cost 50 gm of almonds is $60
Cost 1 gm of almonds is 60/50 = 6/5
By multiplying the above matrices, we get
Cost of pack 1 = 100 + (100)⋅ (1/5) + 50 ⋅ (6/5)
= 100 + 20 + 60
= 180
Cost of pack 2 = 200 + (100)⋅ (1/5) + 100 ⋅ (6/5)
= 200 + 20 + 120
= 340
Cost of pack 2 = 250 + (250)⋅ (1/5) + 150 ⋅ (6/5)
= 250 + 50 + 180
= 480
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