Contents In Matrices

The Matrix plays a vital role in the various computations of mathematics. The properties of matrices are widely used to derive various derivations and formulas. Along with this, one can perform all the Matrix Operations very easily. Matrix Operations include all types of operations like subtraction, addition, Scalar multiplication of constant, and multiplication. Along with this, the Rank of a Matrix and Special Matrices are also studied under matrices. One can also slice the linear equations using the fundamentals of the matrices. We will discuss all these relevant topics in the article. 

Matrix Operations 

All types of operations can be performed using Matrix. Here are some matrix operations listed:

• Addition of two matrices 

• Subtraction between two matrices 

• Multiplication of two matrices 

• Scalar multiplication of two matrices 

Addition of two matrices 

The addition matrix operations are similar to the addition of positive and negative numbers. The additional matrix operations can be performed only when both the matrices consist of equal numbers of rows and columns. In addition to matrix operations, the first Matrix element is added to the corresponding element of the other Matrix. The number of rows and columns in the resultant Matrix will be the same as in the matrices(which are added). 

Subtraction between two matrices 

The subtraction matrix operations are similar to the addition of matrices. Only the difference is that, instead of addition, the two matrices get subtracted. While subtracting the matrices, the elements of the first Matrix get subtracted from the elements of the second Matrix. Likewise, in addition to matrix operation, the number of rows and columns in the subtracting matrices must be the same, and they will also remain the same in the resultant Matrix. 

Multiplication of two matrices 

Multiplication matrix operations are slightly complex in comparison to addition and subtraction. The number of rows in the first Matrix must equal the number of columns in the second Matrix. In the multiplication of two matrices, the elements of rows of the first Matrix get multiplied with the elements of columns. Firstly, the first row is multiplied by the first column, then the second by the second column, and so forth. If the number column is one in the second Matrix, then all the rows of the first Matrix will multiply with the same column. 

Scalar multiplication of two matrices

In the scalar multiplication matrix operations, the given Matrix is multiplied with the constant value k. In scalar multiplication, all the elements of rows and columns of the Matrix get multiplied with the given constant separately. The number of rows and columns in the resultant Matrix remains the same as in the multiplying Matrix. 

The rank of a Matrix and Special Matrices

Let’s understand the Rank of a Matrix and Special Matrices. 

Rank of Matrix 

The rank of a matrix is the highest quantity of rows and columns present in the given Matrix. The rank of the Matrix demonstrates the dimension of the Matrix in the vector space, generally acquired from the number of columns in a matrix. If the number of columns is less than the number of rows, then the rank of the Matrix was acquired from the number of columns in a matrix. 

Special Matrices

There are various types of matrices that have different properties and functionality. Some matrices are called special matrices because they have some unique properties. These matrices work as a special function or property in solving complex problems. There are five types of special matrices. These are:

• Square Matrix 

• Identity matrix 

• Diagonal Matrix 

• Symmetric Matrix 

• Triangular Matrix 

Solving Linear Equations using Matrix

Let’s understand the Solving Linear Equations using Matrix. 

To solve linear equations using Matrix, firstly, all the equations’ variables must be arranged properly. After that, the coefficient of all variables is written in the separate Matrix, and the constant values will be written in the separate Matrix. 

The matrices will be written in the form of:

AX = B

A = Matrix of coefficient of tje equation 

X = Matrix of variables in the equation 

B = Matrix if constant values in the equation 

On performing further computations, 

X = A-1B

A-1 = Transpose of the Matrix. 

So, the solution of linear equations is acquired by obtaining the transpose of the Matrix and multiplying it with the Matrix of constant. So, this is the method for Solving Linear Equations using Matrix.

Conclusion 

Matrix is used to solve many problems in mathematics and other subjects. In simpler words, the Matrix represents numbers in the rows and columns. Matrix Operations include addition, subtraction, and multiplication. The solution or zeros of any equation can be derived easily using the properties of matrices. In solving Linear Equations using Matrix, firstly, one has to derive the transpose of the Matrix, and then to use the transpose, one can derive the solution of the equation. The best thing about this is that zeros of all types of equations, whether binomial or trinomial, can be found using the same method.