In the matrix
write
(i) The
order of the matrix (ii) The number of elements. (iii) Write the elements a13,
a21, a33, a24, a23.
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If a matrix has 24 elements, what are possible orders it can have? What, if it has 13 elements /
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If a matrix has 18 elements, what are possible orders it can have ? What, if it has 5 elements ?
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Construct a 2 × 2
matrix,
whose elements
are given by :
(iii)
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Construct a 3 × 4
matrix, whose elements are given by
(i)
(ii)
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Find the values of x,
y and z from the following equations:
(i)
(ii)
(iii)
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Find the values of a,
b, c and d from the equation
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A = [aij]m × n is a square matrix, if
(a) m < n (b) m > n
(c) m = n (d) none of these
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Which of the given
values of x and y make the following pair of matrices equal
(a)
(b) not
possible to find
(c) y = 7, x =
(d)
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The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is :
(a) 27 (b) 18
(c) 81 (d) 512
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Let
Find each of
the following :
(i)
A + B (ii)
A ? B (iii) 3A ? C (iv) AB (V) BA
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Compute the indicated products :
(i)
(ii)
(iii)
(iv)
(v)
(vi)
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If
then compute (A + B) and (B ? C).
Also, verify that
A + (B ? C) = (A + B) ? C.
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Solve the equation for x, y, z and t.
.
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If
find the
values of x and y.
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If
show that
(x) F(y) = F(x + y).
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If
prove that A3
? 6A2 + 7A + 2I = O.
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If
find K so
that A2 = KA ? 2I.
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IF
is the
identity matrix of order 2, show that
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A trust fund has Rs.30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs.30,000 among the two types of bonds., if the truest fund must obtain an annual total interest of :
(a) Rs.1800 (b) Rs.2000
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The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books. 10 dozen economics books. Their selling prices are Rs. 80 Rs.60 and Rs.40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
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The restriction on n, k and p so that PY + WY will be defined are :
(a) k = 3, p = n
(b) k is arbitrary, p = 2
(c) p is arbitrary, k = 3
(d) k = 2, p = 3.
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If n = p, then the order of the matrix 7X – 5Z is
(a) p × 2 (b) 2 × n
(c) n × 3 (d) p × n
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Find the transpose of
each of the following matrices.
(i)
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If
and
then verify
that
(i) (A + B)?
= A? + B? (ii) (A ? B)? = A? = B?
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If
and
then
verify that
(i) (A + B)? =
A? + B? (ii) (A ? B)? = A? ? B?
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For the matrices A and
B, verify that (AB)? = B?A?, where
(i)
(ii)
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If (i)
then verify
that A?A = I.
(ii)
If
then verify
that A?A = I.
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(i) Show that the matrix
is a symmetric
matrix.
(ii) Show that the matrix
is
as key symmetric matrix.
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For the matrix
verify that
(i) (A + A?) is a symmetric matrix.
(ii) (A ? A)? is a skey symmetric matrix.
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Express
the following matrices as the sum of a symmetric and a skew symmetric matrix :
(i)
(ii)
(iii)
(iv)
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If
then A + A? =
I, if the value of
is
(a)
(b)
(c)
(d)
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Matrices A and B will be inverse of each other only if
(a) AB = BA (b) AB = BA = O (c) AB = O, BA =I (d)
(b) AB = BA = I.
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Let
show
that (aI + bA)n = anI + nan?1 bA, where I is
the identity matrix of order 2 and
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If
, then prove
that An =
where
n is any positive integer.
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If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
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Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
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Find
the values of x, y, z if the matrix
satisfy
the equation A?A = I.
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If
show that A2
? 5A + 7I = O.
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A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below :
Market Products
I 10,000 2,000 18,000
II 6,000 20,000 8,000
(a) If unit sale prices of x, y and z are Rs.2.50, Re.1.50 and Re.1.00, respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Re.2.00, Re.1.00 and 50 paise respectively, find the gross profit.
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If A and B are
square matrices of the same order such that AB = BA, then prove by induction
that ABn = BnA. Further prove that (AB)n = AnBn
for all
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If
is such that
A2 = I, then
(a)
(b)
(c)
(d)
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If the matrix A is both symmetric and skew symmetric, then
(a) A is a diagonal matrix
(b) A is a zero matrix
(c) A is a square matrix
(d) None of these
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If a is square matrix such that A2 = A, then (I + A)3 – 7A is equal to
(a) A (b) I – A
(c) I (d) 3A
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