If
then show
that |2A| = 4 |A|.
If
then show
that |3A| = 27 |A|.
Evaluate the following determinants :
(i)
(ii)
(iii)
(iv)
If
then x is
equal to
(a) 6 (b) ± 6
(c) ?6 (d) 0
Without expanding, prove that
Without expanding, prove that
Without expanding, prove that
Without expanding, prove that
Without expanding, prove that
Without expanding, prove that
Without expanding,
prove that
that
= (a ? b)
(b ? c) (c ? a) (a + b + c)
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Show that
= (x ? y) (y ?
z) (z ? x) (xy + yz + zx)
Let A be a square
matrix of order 3 × 3, then |KA| is equal to
(a) K |A| (b)
K2 |A| (c) K3 (d) 3K |A|
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Which of the following is correct.
(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of these.
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Find the area of the triangle with vertices at the point given is each of the following :
(i) (1, 0), (6, 0), (4, 3) (ii)(2, 7), (1, 1), (10, 8)
(iii) (–2, –3), (3, 2), (–1, –8).
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Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.
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Find the values of k if area of triangle is 4 sq. units and vertices are
(i) (k, 0) (4, 0), (0, 2)
(ii) (–2, 0), (0, 4), (0, k)
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Find the equation of line joining A(1, 2) and B(3, 6) using determinants.
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Find the equation of line joining A(3, 1), and B(9, 3) using determinants.
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If area of tiangle is 35 sq. untis with vertices (2, –6), (5, 4) and (k, 4). Then k is
(a) 12 (b) –2
(c) –12, –2 (d) 12, –2
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Write
minors and cofactors of the elements of following determinants :
(i)
(ii)
Write
minors and cofactors of the following determinants:
(i)
(ii)
Using
cofactors of elements of second row, evaluate
Using
cofactors of elements of third column, evaluate
If
and Aij
is cofactor of aij, then value of
is given
by
(a)
a11 A31
+ a12 A32 + a13A33
(b)
a11A11+
a12A21 + a13A31
(c)
a21 A11
+ a22 A12 + a23 A13
(d)
a11 A11
+ a21 A21 + a31 A31
Find the adjoint of
the matrix
Find the adjoint of
the matrix
Verify A(adj A) = (Adj
A) A = |A| I for matrix
.
Verify A(adj A) = (adj A) = |A| I for matrix
Find the inverse of
the matrix
Find the inverse of
the matrix
Find the inverse of
the matrix
Find the inverse of
the matrix
Find the inverse of
the matrix
.
Let
, verify
that (AB)?1
= B?1
A?1.
If
show
that A2 ? 5A +| 7I = 0. Hence find A?1.
For the matrix
find the
numbers a and b such that A2 + aA + bI = 0.
For the matrix
show
that
A3
? 6A2 + 5A + 11I = 0. Hence find A?1.
If
verify
that A3 ? 6A2 + 9A ? 4I = 0 and hence find A?1.
Let A be a
non-singular matrix of order 3 × 3. Then
|adj A| is equal to
(a) |A| (b)
|A|2
(c) |A|3 (d)
3|A|
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If A is an invertible
matrix of order 2, then det (A?1) is equal to
(a) det
(A) (b)
(c) 1 (d)
0
Examination the consistency of system of equations
X = 2y = 2, 2x + 3y = 3
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Examine the consistency of system of equation
2x – y = 5; x + y = 0
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Examine the consistency of the system of equations
X = 3y = 5; 2x + 6y = 8
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Examine the consistency of the system of equations
x + y + z = 1; 2x + 3y + 2z = 2;
ax + ay + 2az = 4
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Examine the consistency of the system of equations
3x – y – 2z = 2; 2y – z = –1;
3x – 5y = 3
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Examine the consistency of the system of equations.
5x – y + 4 z = 5; 2x + 3y + 5z = 2;
5x – 2y + 6z = –1.
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Solve the following system of linear equations using matrix method
5x + 2y = 4; 7x + 3y = 5
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Solve system of linear equations by matrix method
2x – y = –2; 3x + 4y = 3
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Solve the system of equations using matrix method :
4x – 3y = 3; 3x – 5y = 7
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Solve the system of equations using matrix method
5x + 2y = 3’; 3x + 2y = 5
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Solve the system of
equations using matrix method
2x + y + z = 1;
x ? 2y ? z =
Solve the system of equations using matrix method
x – y + z = 4
2x + y – 3z = 0
x + y + z = 2
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Solve the system of equations using matrix method
2x + 3y + 3z = 5; x – 2y + z = –4; 3x – y – 2y = 3,
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Solve the system of equations using matrix method
x – y + 2z = 7; 3x + 4y – 5z = –5; 2x – y + 3z = 12.
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If A
find A?1.
Using A?1
solve the system of equations.
2x ? 3y + 5z =
11; 3x + 2y ? 4z = ?5; xs + y ? 2z = ?3
The cost of 4 kg onion, 3kg wheat and 2kg rice is Rs60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs.90. The cost of 6kg onion 2kg wheat and 43kg rice is Rs.70. Find cost of each item per kg by matrix method.
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Prove that the
determinant
is
independent of
.
Without expanding,
prove that
If a, b and c are real
numbers, and
=0
show that either a+b+c=0 or a=b=c.
Let
verify
that
(i) [adj
A]?1 = adj (A?1)
(ii) (A?1)?1
= A
Using properties of
determinants, prove that
Using properties of
determinant, prove that
(z ? x),
where p is any scalar.
. Using
properties of determinant, prove that
= 3(a + b + c) (ab + bc + ca)
Using properties of
determinants, prove that
Using properties of
determinant, prove that
If a, b, c are in
A.P., then the determinant
is :
(a) 0 (b)
1
(c) x (d)
2x
If x, y, z are non
zero real numbers, then the inverse of matrix
is
(a)
(b)
(c)
(d)
Let
where
, then
(a) Det (A) = 0 (b)
Det (a)
(c) Det (A)
(d)
Det (A)
(ii) 
find |A|,
= (1 + a2
+ b2)3.
= 1+ a2
+ b2 + c2.
.
and
find
(AB)?1.
