2.Write an example for a row matrix,column matrix, diagonal matrix and zero matrix.
Answers
Step-by-step explanation:
Row Matrix
A row matrix has only one row but any number of columns. A matrix is said to be a row matrix if it has only one row. For example,
A
=
[
−
1
/
2
√
5
2
3
]
is a row matrix of order 1 × 4. In general, A = [aij]1 × n is a row matrix of order 1 × n.
2) Column Matrix
A column matrix has only one column but any number of rows. A matrix is said to be a column matrix if it has only one column. For example,
A
=
⎡
⎢
⎢
⎢
⎢
⎣
0
√
3
−
1
1
/
2
⎤
⎥
⎥
⎥
⎥
⎦
is a column matrix of order 4 × 1. In general, B = [bij]m × 1 is a column matrix of order m × 1.
3) Square Matrix
A square matrix has the number of columns equal to the number of rows. A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example,
A
=
⎡
⎢
⎣
3
−
1
0
3
/
2
√
3
/
2
1
4
3
−
1
⎤
⎥
⎦Rectangular Matrix
A matrix is said to be a rectangular matrix if the number of rows is not equal to the number of columns. For example,
A
=
⎡
⎢
⎢
⎢
⎢
⎣
3
−
1
0
3
/
2
√
3
/
2
1
4
3
−
1
7
/
2
2
−
5
⎤
⎥
⎥
⎥
⎥
⎦
is a matrix of the order 4 × 3
5) Diagonal matrix
A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non-diagonal elements are zero, that is a matrix B =[bij]m×m is said to be a diagonal matrix if bij = 0, when i ≠ j. For example,
A
=
[
4
]
[
−
1
0
0
2
]
⎡
⎢
⎣
3
0
0
0
−
5
0
0
0
2
⎤
⎥
⎦
are diagonal matrices of order 1, 2, 3, respectively.
6) Scalar Matrix
A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant. A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [bij]n × n is said to be a scalar matrix if
bij = 0, when i ≠ j
bij = k, when i = j, for some constant k.
For example,
A
=
[
4
]
[
−
1
0
0
−
1
]
⎡
⎢
⎣
3
0
0
0
3
0
0
0
3
⎤
⎥
⎦
are scalar matrices of order 1, 2 and 3, respectively.
7) Zero or Null Matrix
A matrix is said to be zero matrix or null matrix if all its elements are zero.
For Example,
A
=
[
0
]
[
0
0
0
0
]
⎡
⎢
⎣
0
0
0
0
0
0
0
0
0
⎤
⎥
⎦
are all zero matrices of the order 1, 2 and 3 respectively. We denote zero matrix by O.
8) Unit or Identity Matrix
If a square matrix has all elements 0 and each diagonal elements are non-zero, it is called identity matrix and denoted by I.
Equal Matrices: Two matrices are said to be equal if they are of the same order and if their corresponding elements are equal to the square matrix A = [aij]n × n is an identity matrix if
aij = 1 if i = j
aij = 0 if i ≠ j
We denote the identity matrix of order n by In. When the order is clear from the context, we simply write it as I. For example,
A
=
[
1
]
[
1
0
0
1
]
⎡
⎢
⎣
1
0
0
0
1
0
0
0
1
⎤
⎥
⎦
are identity matrices of order 1, 2 and 3, respectively. Observe that a scalar matrix is an identity matrix when k = 1. But every identity matrix is clearly a scalar matrix.
is a square matrix of order 3. In general, A = [aij] m × m is a square matrix of order m.
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