(1) Construct an m × n matrix A = [a_{ij}], where a_{ij} is given by
(i) a_{ij} = (i - 2j)^{2}/2 with m = 2 and n = 3 Solution
(ii) a_{ij} = |3i - 4j|/4 with m = 3 and n = 4 Solution
(2) Find the values of p, q, r, and s if
(3) Determine the value of x + y if
(4) Determine the matrices A and B if they satisfy
(5) If A =
then compute A^{4 }Solution
(6) Consider the matrix
(i) Show that A_{α} A_{β} = A_{(α+β)}
(ii) Find all possible real values of α satisfying the condition A_{α + }A_{α}^{T} = I Solution
(7) If A =
and such that (A− 2I)(A− 3I) = O, find the value of x.
(8) If A =
(9) If A =
and A^{3} - 6A^{2} + 7A + KI = 0, then find the value of k.
(10) Give your own examples of matrices satisfying the following conditions in each case:
(i) A and B such that AB ≠ BA .
(ii) A and B such that AB = O = BA, A ≠ O and B ≠ O.
(iii) A and B such that AB = O and BA ≠ O.
(11) Show that f (x) f ( y) = f (x + y), where f(x) =
(12) If A is a square matrix such that A^{2} = A, find the value of 7A - (I + A)^{3}. Solution
(13) Verify the property A(B + C) = AB + AC, when the matrices A, B, and C are given by
(14) Find the matrix A which satisfies the matrix relation
(15)
verify the following (i) (A+ B)^{T} = A^{T} + B^{T} = B^{T} + A^{T}
(ii) (A− B)^{T} = A^{T} − B^{T}
(iii) (B^{T} )^{T} = B .
(16) If A is a 3 × 4 matrix and B is a matrix such that both A^{T} B and BA^{T} are defined, what is the order of the matrix B? Solution
(17) Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
(18) Find the matrix A such that
(19) If A
is a matrix such that AA^{T} = 9I , find the values of x and y. Solution
(20) (i) For what value of x, the matrix
is skew-symmetric Solution
(ii) If A =
is skew-symmetric, find the values of p, q, and r.
(21) Construct the matrix A = [a_{ij}]_{3x3}, where a_{ij } = i - j. State whether A is symmetric or skew-symmetric
(22) Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.
(23) 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
(24) 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
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