Identification Of Order, Notation And Types Of Matrices

Identification Of Order, Notation And Types Of Matrices

Types of Matrices

In mathematics, a matrix is a rectangular array of numbers. An item in a matrix is referred as an entry or an element. If the arrangement has m rows and n columns, then the matrix is of order m × n and is read as m by n. There are different types of matrices. They are:

1. Row matrices
2. Column matrices
3. Square matrices
4. Scalar matrices
6. Negative matrices
5. Diagonal matrices
7. Identical matrices

Matrices with only one row is known as row matrices ie A = [a$_{ij}$]_{1 x n}.

$\begin{bmatrix}
0 & 2 &8 & 4 & 3
\end{bmatrix}$ is a 1 x 4 row matrix.

Column Matrices

Matrices with only one column is known as column matrices ie A = [a$_{ij}$]_{m x 1}.

$\begin{bmatrix}
2\\
4\\
6\\
8\\
10
\end{bmatrix}$ is a 5 x 1 column matrix.

Square Matrices

A square matrix has an equivalent number of rows and columns ie A = [a$_{ij}$]_{m x m} .

$\begin{bmatrix}
2 & 7 &8\\
3 & 4 & 5\\
1& 6 & 9
\end{bmatrix}$ is a square matrix having 3 rows and 3 columns.

Scalar Matrices

A diagonal matrix whose diagonal entries is made up of the same elements ie A = [a$_{ij}$]m x m is referred to as scalar matrix if a$_{ij}$ = 0 when i $\neq$ j. a$_{ij}$ = k (constant) when i = j.

$\begin{bmatrix}
c & 0 &0\\
0 & c & 0\\
0 & 0 & c
\end{bmatrix}$ is a scalar matrix.

Diagonal Matrices

A square matrix with all elements not on the key diagonal equal to zero is known as a diagonal matrix. In a matrix A = [a$_{ij}$], an a_ij be the elements in the matrix that is a is element of i^th row and j^th column. If i = j then, it is diagonal elements.

$\begin{bmatrix}
5& 0 &0\\
0 & 6 & 0\\
0 & 0 & 7
\end{bmatrix}$ is a diagonal matrix

Negative Matrices

Matrices of all elements that are by the additive matrices inverse, therefore a matrix A is negative expressed by -A.

If A = $\begin{bmatrix}
3 & 2 &4\\
-14 & 6 & 9
\end{bmatrix}$, then

-A = $\begin{bmatrix}
-3 & -2 &-4\\
14 & -6 & -9
\end{bmatrix}$

And so, A + (-A) = 0

Identical Matrices

An identical matrix is a scalar matrix in which all of the diagonal elements are one or unity.

ie A = [a$_{ij}$]_{m x m} 
if a$_{ij}$ = 1 when i = j and a$_{ij}$ = 0 when i $\neq$ 0.

$\begin{bmatrix}
1 & 0 &0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}$

is an example of identical matrix.

Matrix Notation and Formatting

When you write a matrix, you must make use of squared brackets: " [ ] ". Do not make use of absolute-value bars: " | | ", since they have a different meaning in this context. Do not make use of parentheses or curly braces ( " { } " ) or some other grouping symbol (or no grouping symbol at all), because these presentations have no meaning. A matrix is at all times placed inside square brackets. You must make use of the right notation, or your answers may be counted as incorrect.

As stated earlier, the values that are present inside a matrix are referred to as "entries". For any reason you may have, matrixes are as a custom named with capital letters, like "A" or "B", and the entries are named with the use of the corresponding lower-case letters, but with subscripts. In a matrix A, the entries will normally be named "a_i,j", where "i" is the row of A and "j" is the column of A. For example, given the following matrix A:

...the value 4 is in the second row and the first column, therefore 4 is the 2,1-entry. That is, 4 = a_2,1 (read as "ay-sub-two-one"). The 3,2-entry is the value in the third row and the second column, therefore, a_3,2 = 8. Entry a_1,3 is 3.

For smaller matrices (those with less than ten rows and columns), the comma in the subscript is every so often omitted. For example, "a_1,3 = 3" might be written as "a₁₃= 3". This clearly won't work for larger matrices, since "a₂₁₃" would not be clear. It can mean the 21,3-entry or the 2,13-entry. It is as such very good and recomended, irrespective of the notation used in your book, to use commas in your subscripts, for the sake of clarity.

Further notes on Types of matrices

Sometimes matrices are categorized according to the configurations of their entries.

For example, a matrix with all-zero entries below the top-left-to-lower-right diagonal ("the diagonal") is known as "upper triangular". You can as well have lower triangular matrices, but they aren't of significant use, so "triangular matrice", without the "upper" or "lower", is commonly taken to mean "upper triangular".

A matrix with non-zero entries alone on the diagonal is known as "diagonal".

A diagonal matrix whose non-zero entries are all 1's is known as an "identity" matrix, for reasons which will become clear when you learn how to multiply matrices.

There are a lot of identity matrices. There is the 3 × 3 identity; there is the 4 × 4 identity.

The 3 × 3 identity is represented by I₃ and are read as "eye-three" or "eye-sub-three"; In the same way, the 4 × 4 identity is I₄ and the 2 × 2 identity matrix is I₂:

Observe that triangular matrices are square, that diagonals are triangular and thus are square, and that identities are diagonals and thus are triangular and square. When describing a matrix, you normally just offer it most specific classification, as this implies all the others.

In a triangular matrix, you can have additional zeroes on or above the diagonal.

Classify the following matrix:

This is a diagonal matrix, and, more than that, the diagonal entries are all 1's. Then this is...

the 3 × 3 identity, I₃.Due to the fact that identity matrices are, by definition, square matrices, you are merely required to make use of just one subscript to specify their dimensions.

Matrix Equality

For two matrices to be equivalent, they ought to be of the same size and possess all the same entries in the same places.

These matrices cannot be equivalent due to the fact that they are of different sizes. Even if A and B are the following two matrices:

...they are still not equivalent, although, A and B each have six entries, and the entries are even the same numbers, but that is not all there is for matrices. A is a 3 × 2 matrix and B is a 2 × 3 matrix, and, for matrices, 3 × 2 does not equal 2 × 3. This rule still holds even if A and B have equivalent number of entries or even identical numbers as entries. There can only be equal if A and B are the same size and the same shape and are made up of the same values in precisely the same places, they are not equal.

This property of matrix equality can be used in exam, test or homework questions. You will be provided with two matrices, and told that they are equal. You will be required to make use of this equality to solve for the values of variables.

When carrying out addition, you add each number in the first matrix to the equivalent number in the second matrix.
When doing subtraction, just subtract a number in one of the matrices from the equivalent number in the other matrix.
Addition and subtraction of matrices necessitates that the matrices be the same dimensions. Again, you ought to start and close with the same dimensions.
Scalar multiplication of a real Euclidean vector by a positive real number multiplies the amount of the vector without altering its direction.

A scaler is a quantity that has magnitude but not direction in respect to vector which has both magnitude and direction. 
There are a lot of operations that can be used to alter matrices, like matrix addition and subtraction and scalar multiplication. These form the basic techniques to take care of matrices.

Addition and Subtraction of Matrices

Addition of matrices is very easy. You merely add every number in the first matrix to the equivalent number in the second matrix.

$\begin {pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}+\begin{pmatrix} 10 & 20 & 30 \\ 40 & 50 & 60 \end{pmatrix}=\begin {pmatrix} 11 & 22 & 33 \\ 44 & 55 & 66 \end {pmatrix}$

For example, you can take each number that appears in the upper-right hand corner to produce the calculation 3+30=33. Observe that both matrices being added are 2×3, and the resulting matrix is as well 2×3. You cannot add two matrices that are made up of varied dimensions.

As you may have supposed, subtraction of matrices is similar to the process of addition of matrices apart from the fact that you subtract rather than add.

$\begin{pmatrix} 10 & 20 & 30 \\ 40 & 50 & 60 \end{pmatrix}-\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}=\begin{pmatrix} 9 & 18 & 27 \\ 36 & 45 & 54 \end{pmatrix}$

Again, take note of the fact that the resulting matrix has the same dimensions as the originals, and that you cannot subtract two matrices that have different dimensions.

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