Maharashtra State BoardHSC Science (General) 11th
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Types of Matrices

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Column matrix, Row matrix, Square matrix, Diagonal matrix, Scalar matrix, Identity matrix, Zero matrix

notes

Column matrix:
A matrix is said to be a column matrix if it has only one column.
For example,  A = `[(0), (sqrt 3),(-1),(1/2)]` is a column matrix of order 4 x 1.
In general, A = `[a_(ij)]_(m × 1)` is a column matrix of order m × 1. 

Row matrix:
A matrix is said to be a row matrix if it has only one row.
For example, B = `[-1/2,sqrt 5,2,3]_(1 xx 4)` is a row matrix.
In general, B =`[b_(ij)]_(1 xx n)` is a row matrix of order 1 x n .

Square matrix: 
A matrix in which the number of rows are 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),(3/2,3sqrt2,1),(4,3,-1)]` is a square matrix of order 3.
In general , A = `[a_(ij)]_(m xx m)` is a square matrix of order m . 

Diagonal matrix:
A square matrix B = `[b_(ij)]_(m × m)` is said to be a diagonal matrix if all its non diagonal elements are zero, that is a matrix B = `[b_(ij)]_(m × m)` is said to be a diagonal matrix if `b_(ij)` = 0, when i ≠ j.
For example , A =[4] , B = `[(-1 ,0),(0,2)]` , C = `[(-1.1,0,0),(0,2,0),(0,0,3)],` are diagonal matrices of order 1,2,3,respectively.

Scalar matrix:
A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = `[b_(ij)]_(n × n)` is said to be a scalar matrix if `b_(ij)` = 0,   when i  ≠ j `b _(ij)` = k,   when i  = j, for some constant k. For example
A = [3] , B =` [(-1,0),(0,-1)] `,
C = `[(sqrt3,0,0),(0,sqrt3,0),(0,0,sqrt3)]`
are scalar matrices of order 1, 2 and 3, respectively.

Identity matrix:
A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. In other words, the square matrix A =`[a_(ij)]_(n × n)`is an 
identity matrix, if `a_(ij)` =


\[ a_{ij} = \begin{cases} 1 & \quad \text{if }  \text{  i = j}\\ 0 & \quad \text{if }  \text{ i ≠ j} \end{cases} \] 

We denote the identity matrix of order n by `(I_n)`. When order is clear from the context, we simply write it as I.
 For example [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.

Zero matrix:
A matrix is said to be zero matrix or null matrix if all its elements are zero.
For example, [0],`[(0,0),(0,0)] , [(0,0,0),(0,0,0)]`  , [(0) ,(0)] 
are all zero matrices. We denote zero matrix by O. Its order will be clear from the context.

Video Tutorials

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Shaalaa.com | Matrices class 12 part 6 (Row and column Matrices)

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Matrices class 12 part 6 (Row and column Matrices) [00:04:07]
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