Matrices: Types, Properties, Formulas & Examples

Matrices is defined as the arrangement of numbers, expressions or symbols formed in a rectangular shape. It is organized in the form of rows and columns, which are arranged in horizontal and vertical forms. The total size of the matrix is determined by the number of rows and columns. 

• Matrices is a method by which we can easily solve some complex expressions.
• It helps us study a circuit's charged resources with voltage, resistance, amper, etc. 
• The concept helps in projecting, operating and studying linear graphs.
• Matrices are also useful in representing quadratic forms. 
• It calculates the coefficient of expressions or equations in a meaningful way.
• Matrices are used in many areas like Magnification, Cost Estimation or in the sales projection.

Read More: Matrix Addition

Key Terms: Matrices, Column Matrices, Row Matrices, Square Matrices, Diagonal Matrices, Zero Matrices, Scalar Matrices, Identity Matrices

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What are Matrices?

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Matrices is defined as a rectangular arrangement of m and n elements. The matrices arrangement consists of a m number of rows and n number of columns enclosed within the square brackets. 

• Matrices are shown within the [] or () clauses.
• It is used in cryptographical techniques.
• The entries in a matrix are called elements.
• Horizontal entries of a matrices are called rows.
• Vertical entries of a matrices are called columns.
• Different operations like addition and subtraction can easily be performed on matrices.

Read More: Upper Triangular Matrix

Types Of Matrix

The video below explains this:

Matrices Detailed Video Explanation:

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Types Of Matrices

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There are many types of matrices which can be used in mathematical calculation. Here are the list of different types of matrices.

Column Matrix 

A matrix with only one column is called a Column Matrix. In general, we can say that in the column matrix number of rows is equal to zero but number of columns is equal to one. 

Row Matrix 

A matrix having only one row is called a row natrix. In general, we can say that in the row matrix number of columns is equal to zero but number of rows is equal to one.

Read More: Matrix Multiplication

Square Matrix 

A matrix having same number of rows and same number of columns is called a Square Matrix. If the size of a Matrix is m*n then in the Square Matrix there is m=n .

Diagonal Matrix 

A matrix that has an element only on diagonal position then it is called a Diagonal Matrix. 

Zero Matrix 

A matrix having a zero on all the positions is called a Zero Matrix.

Scalar Matrix 

A diagonal matrix having the same elements on diagonal Position is called as a Scalar Matrix.

Identity Matrix 

In the square matrix, in which all elements on diagonal positions are equal to one and rest elements are 0 called an Identity Matrix.

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Operation on Matrices 

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The list of operation performed on matrices are as follows:

Addition 

Addition of two matrices means we have to add the elements of one matrix with the other matrix in row wise or column wise. It means we have to do addition of elements. 

Multiplication 

Multiplication of matrices means we have to multiply the elemnets of one matrix with the other matrix in row wise or column wise. It means we have to do multiplication of elements. 

Subtraction 

Subtraction of matrices means we have to subtract the elemnets of one matrix with the other matrix in row wise or column wise. It means we have to do subtraction  of elements. 

Read More: Applications of Determinants and Matrices

Read More: Inverse Matrix Formula

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Properties of Matrices 

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The important properties of matrices are as follows:

• -A = (-1)A
• A – B = A + (-1) x B
• A + B = B + A
• (A + B) + C = A + (B + C), where A,B and C are in the same order of matrix.
• k(A + B) = k x (A) + k x (B), where k is a constant and A,B are in same order of matrix
• ((A)’)’ = A
• (A + B)’ = A’ + B’
• (AB)’ = B’A’

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Important Formulas of Matrices

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The important formulas of matrices are as follows:

• A^-1 = adj(A)/|A|
• |adj A| = |A|n-1 where n is the order of matrix A
• A(adj A) = (adj A)A = I, where I is an Identity Matrix
• adj(adj A) = |A|n-2A where n is the order of the matrix
• adj(AB) = (adj B)(adj A)
• |adj(adj A)| = |A|^{(n-1)^2}
• adj(A^p) = (adj A)^p
• If A is symmetric then adj(A) is also symmetric
• adj(kA) = kn-1(adj A) where k is any real number
• adj(I) = I
• If A is a diagonal Matrix then adj(A) is also a diagonal matrix
• adj 0 = 0
• If A is a triangular matrix then adj(A) is also a triangular matrix
• If A is a singular Matrix then |adj A| = 0

Read More: Operations on Matrices

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Things to Remember

• Matrices is a rectangular table of numbers, symbols or expressions.
• The elements are arranged in the form of rows and columns.
• It is used to represent linear map and can solve problems on linear algebra.
• Addition, subtraction and multiplication are some of the operations performed on matrices.
• The size of a particular matrix can be defined by the number of rows and columns.

Read More: Transpose of a Matrix

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Previous Years Questions

• If R(t)=[costsint−sintcost], then R(s) R(t) equals...[BITSAT 2017]
• If for a matrix A, |A| = 6 and adj A=⎡⎢⎣1−24411−1k0⎤⎥⎦ , then k is equal to :...[JEE Main 2018]
• If X+Y=[7025] and X−Y=[3003], then X is equal to…..[JEE Main 2013]
• If ∣∣∣x+75x+−33∣∣∣=26, then x is equal to….[JKCET 2013]
• If $2A+3B =[2−14325]andA+2B[503162]thenB =$….[JEE Advanced 2006]
• The only integral root of the equation ∣∣∣∣|2−y2325−y63410−y|=0 is​
• The value of ∣∣ ∣∣b+caabc+abcca+b∣∣ ∣∣|b+caabc+abcca+b| is​
• If the three linear equations x+4ay+az=0x+4ay+az=0 x+3by+bz=0x+3by+bz=0 and x+2cy+cz=0x+2cy+cz=0 have a non-trivial solution, then a, b, c are in
• The matrix 'X' in the equation AX=B, such that A=[1301] and B=[1−101] is given ​
• If ω≠1 is the complex cube root of unity and matrix H=[ω00ω], then H70 is equal to -​
• The number of 3×3 non-singular matrices, with four entries as 1 and all other entries as 0, is​
• IfBistheinverseofA,then.IfBistheinverseofA,then\alpha$ is​
• A square matrix A=[aij]n×n is called a diagonal matrix if aij=0 for​
• A square matrix A is said to be singular if​
• A square matrix A=[aij]n×n is called a lower triangular matrix if aij=0 for