Matrix
Content
MATRICES
1. If a matrix has 24 elements what are the possible orders it can have? What if,
it has 13 elements?
2. Construct a 2X2 matrix A= [aij], whose elements are given by: a a ij = i+2j
3. Construct a 3X2 matrix A= [aij], whose elements are given by:
(i) [aij] = ½ |i-3j|
(ii) [aij] = (i+2j)2 /2
4. Write the value of x-y+z, if matrix
¿
x+ y+ z ¿ y+ z ¿ x+ z ¿ =
¿
¿
9 ¿7 ¿ 5 ¿
¿
] [ ]
] [ ]
]
[
]
[ ] [ ]
5. Find the value of y if
6. Find the value of x if
[
7. If
9 −1 4
−2 1 3
[]
2
3
8. If x
+y
[
[
x−y 2
x
5
3 x + y −y
2 y−x 3
1 2 −1
0 4 9
=A+
−1
1
=
=
2 2
3 5
=
1 2
−5 3
.
, find A
10
5
9. Find the value of x+y from the following matrix equation:
[
]
3 −4 α
1 2
=
[
7 6
15 14
]
10.Find the value of a matrix X such that 2A+B+X=0 where
A=
11. If
[ ]
[ ]
[ ][ ] [ ]
[ ]
−1 2
3 4
B=
1 2 3 1
3 4 2 5
12.If A=
2 3
1 2
=
3 −2
1 5
7 11
k 23
, then write the value of k.
, prove that A3-4A2+A=0
2
[
x
7
5
y−3
]
+
13.If A=
14.If A=
[
[
15.If f(x)=
16.If A=
[
1 0
−1 7
]
]
, such that A2-8A+kI=0
3 4
−4 3
[
, find f(A), where f(x)=x2-5x+7
cosx −sinx 0
sinx cosx 0
0
0
1
cosx sinx
−sinx cosx
]
]
, show that f(x).f(y)=f(x+y)
, prove that An=
[
cosnx sinnx
−sinnx cosnx
]
, for all n.
17.Solve for x,y given that
[
][ ] [ ]
2 −3 x
1
=
1 1 y
3
[
18.Solve for x and y , given that
19.If A=
I)
II)
x
3y
[ ] [ ]
2 3 ∧I = 1 0 ,
then
1 12
0 1
Find α and µ ,so that A2=αA+ µI
Prove that A3-4A2+A=o
20.Find the matrix A such that
[ ]
[
[ ]
1 1
0 1
21.Find a 2x2 matrix B such that B
22.If A=
][ ] [ ]
y 1
3
=
x 2
5
[ ]
4 1
5 8
A=
1 −2
1 4
2 3 3 5
1 0 1
]
=6I.
, show that A+AT is a symmetric matrix, where AT denotes the
transpose of A.
23.If A=
[ ]
1 4
3 7
, show that A- AT is skew symmetric, where AT is the transpose
of A.
24.If the matrix
25.If the matrix
[
0 6−5 x
x2
7
[
−2
1
x+ y
]
is symmetric, find the value of x.
x− y 5
0
4
z
7
]
is symmetric, find the values of x,y and z.\
26. For what values of x, is the matrix A=
[
0
1 −2
−1 0
3
x −3 0
]
27.Find ½ ( A+AT){Symmetric} and ½ (A-AT){Skew-symmetric} where A=
[
0
a b
−a 0 c
−b −c b
]
1. If a matrix has 24 elements what are the possible orders it can have? What if,
it has 13 elements?
2. Construct a 2X2 matrix A= [aij], whose elements are given by: a a ij = i+2j
3. Construct a 3X2 matrix A= [aij], whose elements are given by:
(i) [aij] = ½ |i-3j|
(ii) [aij] = (i+2j)2 /2
4. Write the value of x-y+z, if matrix
¿
x+ y+ z ¿ y+ z ¿ x+ z ¿ =
¿
¿
9 ¿7 ¿ 5 ¿
¿
] [ ]
] [ ]
]
[
]
[ ] [ ]
5. Find the value of y if
6. Find the value of x if
[
7. If
9 −1 4
−2 1 3
[]
2
3
8. If x
+y
[
[
x−y 2
x
5
3 x + y −y
2 y−x 3
1 2 −1
0 4 9
=A+
−1
1
=
=
2 2
3 5
=
1 2
−5 3
.
, find A
10
5
9. Find the value of x+y from the following matrix equation:
[
]
3 −4 α
1 2
=
[
7 6
15 14
]
10.Find the value of a matrix X such that 2A+B+X=0 where
A=
11. If
[ ]
[ ]
[ ][ ] [ ]
[ ]
−1 2
3 4
B=
1 2 3 1
3 4 2 5
12.If A=
2 3
1 2
=
3 −2
1 5
7 11
k 23
, then write the value of k.
, prove that A3-4A2+A=0
2
[
x
7
5
y−3
]
+
13.If A=
14.If A=
[
[
15.If f(x)=
16.If A=
[
1 0
−1 7
]
]
, such that A2-8A+kI=0
3 4
−4 3
[
, find f(A), where f(x)=x2-5x+7
cosx −sinx 0
sinx cosx 0
0
0
1
cosx sinx
−sinx cosx
]
]
, show that f(x).f(y)=f(x+y)
, prove that An=
[
cosnx sinnx
−sinnx cosnx
]
, for all n.
17.Solve for x,y given that
[
][ ] [ ]
2 −3 x
1
=
1 1 y
3
[
18.Solve for x and y , given that
19.If A=
I)
II)
x
3y
[ ] [ ]
2 3 ∧I = 1 0 ,
then
1 12
0 1
Find α and µ ,so that A2=αA+ µI
Prove that A3-4A2+A=o
20.Find the matrix A such that
[ ]
[
[ ]
1 1
0 1
21.Find a 2x2 matrix B such that B
22.If A=
][ ] [ ]
y 1
3
=
x 2
5
[ ]
4 1
5 8
A=
1 −2
1 4
2 3 3 5
1 0 1
]
=6I.
, show that A+AT is a symmetric matrix, where AT denotes the
transpose of A.
23.If A=
[ ]
1 4
3 7
, show that A- AT is skew symmetric, where AT is the transpose
of A.
24.If the matrix
25.If the matrix
[
0 6−5 x
x2
7
[
−2
1
x+ y
]
is symmetric, find the value of x.
x− y 5
0
4
z
7
]
is symmetric, find the values of x,y and z.\
26. For what values of x, is the matrix A=
[
0
1 −2
−1 0
3
x −3 0
]
27.Find ½ ( A+AT){Symmetric} and ½ (A-AT){Skew-symmetric} where A=
[
0
a b
−a 0 c
−b −c b
]
