1.Algebra Booster

7.10 Algebra Booster
10. Unitary matrix
A square matrix A is called unitary, if AA q = I.
1 Ê 1 1+
iˆ
For example, A = Á
3 Ë1-i
-1˜
¯
Note If A and B are unitary, then AB is also a unitary.
11. Equivalent matrices
Let A and B are two matrices. If B is obtained from A by elementary
transformation, then A and B are called equivalent
matrices.
12. Rank of a matrix
Rank of a matrix represents the non-zero rows of an equivalent
matrix.
Rule to find out the rank of a matrix
Let A be any type of matrix
Case I: When A is a null matrix, then the rank of a matrix
is zero.
Case II: When A is a square matrix, then we shall first find
the determinant of A.
(i) If A is non-singular ( i.e. |A| π 0), then rank of the matrix
= order of the matrix
(ii) If A is singular (i.e. |A| = 0), then we shall find the minor
along rows:
(a) If at-least one minor is zero and rest are non-zero,
then rank of the matrix = order of the matrix –1.
(b) If all minor is non-zero, then rank of the matrix =
0.
Case III: When A is a rectangular matrix of order m ¥ n,
then we shall find an equivalent matrix of A.
(i) If any one row is zero, then rank of the matrix = Minimum
of {m – 1, n – 1}
(ii) If any two row is zero, then rank of the matrix = Minimum
of {m – 2, n – 2 }
(iii) If all rows are non-zero, then rank of the matrix = Minimum
of {m, n}.
EXERCISES
LEVEL I
ORDER OF MATRICES
(Problems based on Fundamentals)
1. Find the number of all possible matrices of order 2 ¥ 2
with each entry 0 or 1.
2. Find the number of all possible matrices of order 3 ¥ 3
with each entry either 1 or 2.
ADDITION OF MATRICES
2 4
3. If A = Ê ˆ
Á
Ë3 5 ˜ , find the additive inverse of A.
¯
2 5 3 6
4. Find a matrix X, if X + Ê ˆ Ê ˆ
Á = .
Ë3 -2˜ ¯
Á
Ë2 7˜
¯
5. Find X and Y, if
Ê4 2 ˆ
and X - Y = Á
Ë8 -2
˜
¯ .
6. Find a matrix X such that
Ê2 5 ˆ
X + Y = Á .
Ë3 -2
˜
¯
A + 2B + X = O,
Ê2 1 1 1
where A
- ˆ Ê
and B
- ˆ
= Á =
Ë3 5 ˜
¯
Á
Ë 0 2˜
¯ .
7. Find x and y, if
Ê|| x 2 ˆ Ê<
3 2ˆ
Á =
Ë 5 | y - 2| ˜
¯
Á
Ë 5 < 4˜
¯
8. Find S ( x + y)
, if
Ê
3
x - 3x
+ 2 2 ˆ Ê0 2ˆ
Á
3 2
˜ = Á
3 y 7y
35 Ë3 1˜
Ë
+ - ¯ ¯
9. Find x, y, z and t satisfying the equations
Êx
yˆ Ê1 -2ˆ Ê3 5ˆ
2Á + 3 = 4
Ëz
t˜ ¯
Á
Ë0 4 ˜
¯
Á
Ë4 6˜
¯
10. Find the matrices X and Y, if
Ê2 3ˆ Ê-1 2 ˆ
2X + 3Y = Á and3X + 2Y
=
Ë4 0˜ ¯
Á
Ë 1 -5˜
¯
Ê2 0ˆ Ê1 0ˆ
11. If A= Á and I = ,
Ë0 2˜ ¯
Á
Ë0 1˜
¯
and f(x) = 1 + x + x 2
f(A).
MULTIPLICATION OF MATRICES
12. Let A= [1 2] and B =
Find AB and BA.
È2 ˘
Í .
3
˙
Î ˚
È
2
a ˘
Í ˙
2
13. Let A= [a bc] and B = Íb
˙.
2
ÍÎc
˙˚
Find AB and BA.
1 2
14. Let A = Ê ˆ
Á
Ë3 4 ˜
¯ and 2 4
B = Ê ˆ
Á .
Ë5 7 ˜
¯
Find AB and BA.