Types and Algebra of Matrices

Types of Matrix: Matrices are basically a rectangular array of numbers either real or complex which we can denote as follows.

\(A =\begin{bmatrix} {a_1}{_1} & {a_1}{_2} & {a_1}{_3} \\ {a_2}{_1} & {a_2}{_2} &  {a_2}{_3} \\ {a_3}{_1} & {a_3}{_2} & {a_3}{_3} \end{bmatrix} \)

Hence, a matrix has m number of rows and n number of columns. We use brackets i.e. [] or () for denoting matrices. Let’s know more about the types of matrix.

 

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Types of Matrix

There are many types of matrix, some of which are:

I. Row Matrix

A row vector or a row matrix consists of a single row. Some of the examples of row matrices are:

• [a₁ a₂ a₃ ………. a_n]
• [1, 3, 9]

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II. Column Matrix

A column vector or a column matrix consists of a single column. Some of the examples of column matrices are:

• \(\begin{bmatrix} {a_1}{_1} \\ {a_2}{_1} \\ {a_3}{_1}\end{bmatrix} \)
• \(\begin{bmatrix} {1} \\ {2} \\ {3} \end{bmatrix} \)
• \(\begin{bmatrix} {2} \\ {7} \end{bmatrix} \)

III. Zero Matrix or Null Matrix

If a matrix m × n, has all of its elements as zero, then we refer to such a matrix as a zero or a null matrix. A zero or null matrix has an order of (m,n). We can represent a zero or null matrix as O_m×n. Here are some of the examples of zero matrices with different orders:

• \(\begin{bmatrix} {0} & {0} \\ {0} & {0} \end{bmatrix} \)
• \(\begin{bmatrix} {0} \\ {0} \\ {0} \end{bmatrix} \)

IV. Square and Rectangular Matrix

When a matrix has same number of rows and columns, then such a matrix is a square matrix. Furthermore, if that is not the case, then the given matrix is a rectangular matrix. For example,

Square Matrix: \(A =\begin{bmatrix} {1} & {-1} & {9} \\ {2} & {8} & {2} \\ {-3} & {3} & {4} \end{bmatrix} \)

V. Diagonal Matrix

If all the elements of a matrix are zero except for its diagonal elements (top-left to right bottom), then it is a diagonal matrix. Hence, if

\(A =\begin{bmatrix} {a_1}{_1} & {a_1}{_2} & {a_1}{_3} \\ {a_2}{_1} & {a_2}{_2} &  {a_2}{_3} \\ {a_3}{_1} & {a_3}{_2} & {a_3}{_3} \end{bmatrix} \)  = [a_ij]_n×n

It is a n × n matrix. Hence, the diagonal matrix we get is:

Diagonal Matrix: \(A =\begin{bmatrix} {a_1}{_1} & {0} & {0} & {0} \\ {0} & {a_2}{_2} &  {0} & {0} \\{.} & {.} & {.} & {.} \\  {0} & {..} & {..} & {a_m}{_n} \end{bmatrix} \)

VI. Scalar Matrix

When a diagonal matrix has all of its diagonal elements equal to some scalar quantity, say k, then we call it a scalar matrix. For example,

• \(A =\begin{bmatrix} {k} & {0} & {0} \\ {0} & {k} &  {0} \\ {0} & {0} & {k} \end{bmatrix} \)
• \(B =\begin{bmatrix} {1} & {0} \\ {0} & {1} \end{bmatrix} \)

VII. Unit Matrix

Unit matrix is a scalar matrix in which the diagonal elements are 1 or unity. We can denote such a matrix using I_m×n. For example,

\( A =\begin{bmatrix} {1} & {0} & {0} & {0} \\ {0} & {1} &  {0} & {0} \\{.} & {.} & {.} & {.} \\  {0} & {..} & {..} & {1} \end{bmatrix} \) =  I_{m × n}

VIII. Upper Triangle Matrix

When all the elements below the diagonal are null or zero, then such a matrix is upper triangle matrix. For example,

\(A =\begin{bmatrix} {1} & {2} & {3} \\ {0} & {5} &  {6} \\{0} & {0} & {9} \end{bmatrix} \)

IX. Lower Triangle Matrix

When all the elements above the diagonal are null or zero, then such a matrix is lower triangle matrix. For example,

\(A =\begin{bmatrix} {1} & {0} & {0} \\ {4} & {5} &  {0} \\{7} & {8} & {9} \end{bmatrix} \)

X. Sub Matrix

The matrix that we get on deleting one or more than one rows or columns or even both is a sub matrix. For example,

Let \(A =\begin{bmatrix} {1} & {3} & {9} \\ {2} & {9} &  {1} \\{-7} & {3} & {4} \end{bmatrix} \)

Therefore, on deleting the second row and second column from the matrix A, we get:

\(B =\begin{bmatrix} {1} & {9} \\ {-7} & {4} \end{bmatrix} \)

XI. Equal Matrices

Two given matrices, X = [x_ij] and Y = [y_ij] are equal matrices if they are valid for the following conditions.

• The order of X and Y is the same
• The corresponding elements in matrices X and Y are same, i.e. x_ij = y_ij (i = 1,2,3, …..m and j = 1,2, ….n)

That’s all for the types of matrix. Now, let’s take the discussion further and learn about the algebra of matrices in the following section. Here, we will try to understand how do we operate on these different types of matrix.

Algebra of Matrices

Source: math

I. Addition and Subtraction

Take two matrices, X and Y in the same order. Now, the addition of these two matrices is X + Y. Similarly, to subtract the matrices, we simply subtract the corresponding elements. Hence, if  X = (x_ij)_{m × n} and Y = (y_ij)_{m × n}, then

X + Y = (x_ij + y_ij)_{m × n}

Property

If three matrix, X Y, and Z are of the same order. Then,

• Commutative Law: X + Y = Y + X
• Associative Law: (X + Y) + Z = X + (Y + Z)
• K (X + Y) = k.A + k.B. Here, k = constant

Example

\(\begin{bmatrix} {1} & {-1} & {9} \\ {2} & {8} & {2} \\ {-3} & {3} & {4} \end{bmatrix} \) + \(\begin{bmatrix} {1} & {-1} & {9} \\ {2} & {8} & {2} \\ {-3} & {3} & {4} \end{bmatrix} \) =  \(\begin{bmatrix} {2} & {-2} & {18} \\ {4} & {16} & {4} \\ {-6} & {6} & {8} \end{bmatrix} \)

II. Negative of a Matrix

We get the negative of any matrix by changing the signs of all of its elements.

Given Matrix : \(A =\begin{bmatrix} {4} & {2} & {1} \\ {2} & {-8} & {2} \\ {-3} & {-1} & {-7} \end{bmatrix} \)

Negative of the Matrix: \(A = \begin{bmatrix} {-4} & {-2} & {-1} \\ {-2} & {8} & {-2} \\ {3} & {1} & {7} \end{bmatrix} \)

III. Scalar Multiplication

When we take a scalar quantity, say, k, then the multiplication of a matrix by that scalar is:

Example

Let \(B =\begin{bmatrix} {2} & {3} \\ {-7} & {4} \end{bmatrix} \)k = 3

Then, we get,

k B = \(B =\begin{bmatrix} {6} & {9} \\ {-21} & {12} \end{bmatrix} \)

IV. Multiplication

For the multiplication of two matrices, the column in Matrix X must be equal to the number of rows in Matrix Y.

Properties

• Multiplication of matrix is not commutative i.e. XY ≠ YZ
• Multiplication of matrix is associative i.e. (XY)Z = X(YZ)
• X(Y + Z) = XY + XZ. Similarly, (X + Y)Z = XZ + YZ
• If we have X, Y, and Z in such a way that XY = XZ then, Y = Z if X is an invertible matrix
• If we take two non-zero matrices, then the product of those two is non-zero as well

V. Transpose of a Matrix

When we exchange the rows and columns of a particular matrix then we call it a transpose. The representation of a transpose is A^T or A’. Hence,  if , X = [x_ij] and X’ = [y_ij], therefore, x_ij = yji.

Example

\(X =\begin{bmatrix} {1} & {3} & {9} \\ {2} & {1} &  {0} \\{4} & {2} & {1} \end{bmatrix} \)
then,
\(X’ =\begin{bmatrix} {1} & {2} & {4} \\ {3} & {1} &  {2} \\{9} & {0} & {1} \end{bmatrix} \)

Properties

• In general, a matrix is a transpose of its own transpos matrix i.e., X = (X’)’
• The transpose of a matrix follows the equation (X + Y)’ = X’ + Y’
• (XY)’ = Y’X’

VI. Symmetric Matrix

If a given matrix “X” is symmetric, then, X’ = X.

Example

\(X =\begin{bmatrix} {1} & {-1} & {5} \\ {-1} & {2} &  {1} \\{5} & {1} & {3} \end{bmatrix} \).
Then,
\(X’ =\begin{bmatrix} {1} & {-1} & {5} \\ {-1} & {2} &  {1} \\{5} & {1} & {3} \end{bmatrix} \).

In here, X’ = X, where, X is called a symmetric matrix.

VII. Skew Symmetric Matrix

A given matrix “X” is skew-symmetric if X’ = -X. Symbolically,

X =  [x_ij] then, x_ij = -x_ji.

Example

\(X =\begin{bmatrix} {0} & {-5} & {4} \\ {5} & {0} &  {-1} \\{-4} & {1} & {0} \end{bmatrix} \).
Then,
\(X’ =\begin{bmatrix} {0} & {5} & {-4} \\ {-5} & {0} &  {1} \\{4} & {-1} & {0} \end{bmatrix} \).

In here, X’ = -X. So, X is called a skew-symmetric matrix.

Solved Example on Types and Algebra of Matrices

Question: Find X².  Given \(X =\begin{bmatrix} {1} & {0} & {0} \\ {0} & {1} &  {0} \\{a} & {b} & {-1} \end{bmatrix} \)

Solution:

X² = X.X = \(\begin{bmatrix} {1} & {0} & {0} \\ {0} & {1} &  {0} \\{a} & {b} & {-1} \end{bmatrix} \)\(\begin{bmatrix} {1} & {0} & {0} \\ {0} & {1} &  {0} \\{a} & {b} & {-1} \end{bmatrix} \)

= \(\begin{bmatrix} {1×1+0×0+0×a} & {1×0+0×1+0×b} & {1×0+0×0+0×1} \\ {0×1+1×0+0×a} & {0×0+1×1+0×b} &  {0×0+1×0+0×-1} \\{a×1+b×0-1×a} & {a×0+b×1+(-1)×b} & {a×0+b×0+(-1)×(-1)} \end{bmatrix} \)

= \(\begin{bmatrix} {1} & {0} & {0} \\ {0} & {1} &  {0} \\{0} & {0} & {1} \end{bmatrix} \) = 1

This concludes our discussion on the topic of types of matrix.