Matrices – Complete Guide For Class 12 Math Chapter 3

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The chapter Matrices introduces students to matrices, which are essential for solving systems of linear equations and performing various mathematical operations. It covers different types of matrices, including row, column, square, diagonal, scalar, identity, and zero matrices. Key operations such as addition, subtraction, and multiplication of matrices are discussed, along with the concept of the determinant, which helps in finding matrix inverses. The inverse of a matrix, when it exists, is crucial for solving matrix equations using methods like Cramer’s rule and matrix inversion. Matrices are foundational for higher studies in mathematics, engineering, and computer science, with applications extending to fields such as linear programming, computer graphics, and data analysis.

Matrices

Matrix: A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, it is called an m × n matrix.

For example:  A   =  

 Is a 3 × 2 matrix.

Understanding Matrices Through Examples

Consider the following table representing the number of notebooks and pens three students have:

This information can be represented as a matrix:  

                  A   =                        

Order of a Matrix

The order of a matrix is determined by the number of rows and columns it contains. 

For example: A    =

is a matrix of order 2 × 3.

Consider the following data of male and female students in three classes:

This can be represented as a matrix:

        B     = 

The entry in the first row and second column (25) represents the number of female students in Class I.

Constructing Matrices

Let’s construct a 2 × 2 matrix where the elements are given by aᵢⱼ = i + 2j:

          A   =     

  Where:

(i) a₁₁ = 1 + 2 (1) = 3

(ii) a₁₂ = 1 + 2 (2) = 5

(iii) a₂₁ = 2 + 2 (1) = 4

(iv)a₂₂  = 2 + 2 (2) = 6

Types of Matrices

• Column Matrix: A matrix with only one column.

                            A      =   ​

                 is a column matrix of order 3 × 1.

• Row Matrix: A matrix with only one row.

                                    B     =

                   is a row matrix.

• Square Matrix: A matrix where the number of rows equals the number of columns.

                                        C     =                

                                    is a square matrix of order 3.

• Diagonal Matrix: A square matrix where all non-diagonal elements are zero.

                                    D  = 

                           is a diagonal matrix of order 2.

• Scalar Matrix: A diagonal matrix where all diagonal elements are equal.

                              E     =                       

                          is a scalar matrix of order 3.

• Identity Matrix: A square matrix where all diagonal elements are 1, and all other elements are 0.  

        I     = 

​

                Is an identity matrix of order 3.

• Zero Matrix: A matrix where all elements are zero.

                          O     =  

                          is a zero matrix of order 2.

Equality of Matrices

Two matrices are equal if:

• They have the same number of rows and columns.
• Their corresponding elements are equal.

For example:

Given:

                    =                                      

We find:

(i) x + 4 = 0 ⟹ x =−4

(ii) y + 2 = 5 ⟹ y = 3

(iii) z − 1 = −2 ⟹ z =−1

(iv) 2a =−6 ⟹ a =−3

Operations on Matrices

1. Matrix Addition

For matrices A and B of the same order: 

A + B =

Properties of Matrix Addition:

• Commutative: If A =[aᵢⱼ] and B= [bᵢⱼ] are two matrices of the same order, then

                    A + B = B + A

• Associative: If A =[aᵢⱼ], B= [bᵢⱼ] and c= [cᵢⱼ] are any three matrices of the same order, then

                    (A + B) + C = A + (B + C)

• Additive Identity: If A is an m × n matrix and O is an m × n zero matrix, then

                    A + O = O + A = A

• Additive Inverse: If A is an m × n matrix and (-A) is the negative of A, then the addition of matrix with the negative of itself is always a null matrix. Here (-A) is the additive inverse of A. 

                    A + (−A) = (−A) + A = O

2. Matrix Subtraction

For matrices A and B of the same order: A – B = 

​3. Multiplication of a Matrix by a Scalar

For a matrix A and scalar k: kA =

Properties of Scalar Multiplication

• k (A+B) = kA + kB
• (k₁​+k₂​)A=k₁​A+k₂A

Matrix Multiplication

Matrix multiplication is a key operation in matrix algebra. If we have two matrices, A of order m×n and B of order n×p, their product AB is a matrix of order m×p. The element in the i-th row and j-th column of AB is obtained by taking the dot product of the i-th row of A with the j-th column of B.

Example: Consider matrices A =

and  B  =

To find AB, we calculate:

                  AB   =   x

 =

= 

Properties of Matrix Multiplication

• Non-Commutativity: Matrix multiplication is generally not commutative, meaning AB ≠ BA in most cases.

• Associativity: Matrix multiplication is associative. For any three matrices A, B, and C, A(BC)=(AB)C.
• Distributivity: Matrix multiplication is distributive over addition. For matrices A, B, and C, we have: 

 A(B + C) = AB + AC and (A + B)C = AC + BC

• Identity Matrix: For every square matrix A, there exists an identity matrix I such that IA = AI = A.
• Null Matrix Product: The product of two matrices can be a null matrix even if neither matrix is null.
• Multiplication with Null Matrix: If A is any matrix and O is a null matrix, then AO or OA is always a null matrix.
• Non-Zero Products: If AB = O, it does not necessarily imply BA=O.

Transpose and Symmetry

• The transpose of a matrix A, denoted as A, is obtained by flipping the matrix over its diagonal.
• A square matrix A is symmetric if A = A 
• A square matrix A is skew-symmetric if A = −A 

Transpose of a Matrix: The transpose of a matrix is obtained by changing the rows into columns and columns into rows. It is denoted by Aᵀ or A’. 

Example:   Transpose of A        = 

is A’      = 

 Properties of Transpose

Symmetric Matrix

A square matrix A = [aᵢⱼ] is called a symmetric matrix if aᵢⱼ = aⱼᵢ for all i, j.

Example: A        = 

Is a symmetric matrix as A’ = A.

Skew-Symmetric Matrix

A square matrix A = [aᵢⱼ] is called a skew-symmetric matrix if aᵢⱼ = – aⱼᵢ for all i, j.

Example: A        = 

Is a symmetric matrix as A’ = A.

Theorem 1: For any square matrix A with real number entries, A+A′ is a symmetric matrix and A – A′ is a skew-symmetric matrix. 

Proof: Let B = A+A′. Then B’ = (A + A′)′ = A′ + (A′)′ ( as (A + B)′ = A′ + B′ ) 

                                                                            = A′ + A ( as (A′)′ = A ) 

                                                                            = A + A′ ( as A + B = B+A ) 

                                                                            = B 

Therefore B = A + A′ is a symmetric matrix.

Now let C = A – A′ 

                C′ = (A – A′)′ 

                     = A′ – (A′)′ 

                     = A′ – A 

                    = – (A – A′) 

                   = – C 

Therefore C = A – A′ is a skew symmetric matrix. 

Now let C = A – A′ C′ = (A – A′)′ = A′ – (A′)′ = A′ – A = – (A – A′) = – C Therefore C = A – A′ is a skew symmetric matrix.

Theorem 2: Any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.             

Proof: Let A be a square matrix, then we can write

A= 1/2(A + A’) + 1/2 (A – A’)

From Theorem 1, we know that A + A’ is a symmetric matrix and A – A’ is a skew-symmetric matrix.

1/2(A + A’) is symmetric matrix and 1/2 (A – A’) is skew-symmetric matrix.   

Thus, any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix. 

Invertible Matrix: A square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = BA = I. 

Example: A    = 

and  B     = 

Be two matrices. AB = 

Theorem 3: (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. 

Proof: Let A = [aᵢⱼ] be a square matrix of order m. If possible, let B and C be two inverses of A. We shall show that B = C. 

Since B is the inverse of A 

AB = BA = I … (1).

Since C is also the inverse of A 

AC = CA = I … (2) 

B = BI = B (AC) ( from (2) )

             = (BA) C ( associative property) 

             = IC ( from (2) ) 

             = C 

Thus B = C. 

Theorem 4: If A and B are invertible matrices of the same order, then (AB)⁻¹ = B⁻¹ A⁻¹. 

Proof: From the definition of the inverse of a matrix, we have (AB) (AB)⁻¹ = I 

or A⁻¹ (AB) (AB)⁻¹ = A⁻¹I (Pre multiplying both sides by A⁻¹) 

or (A⁻¹A) B (AB)⁻¹ = A⁻¹ (Since A⁻¹ I = A⁻¹ )

or IB (AB)⁻¹ = A⁻¹ 

or B (AB)⁻¹ = A⁻¹

or B⁻¹ B (AB)⁻¹ = B⁻¹ A⁻¹ 

or I (AB)⁻¹ = B⁻¹ A⁻¹

Hence (AB)⁻¹ = B⁻¹ A⁻¹

Example: 

Let’s Conclude

In conclusion, Chapter 3 – Matrices is a vital part of the CBSE Class 12th Mathematics syllabus, providing students with essential knowledge and skills for advanced mathematical applications. By mastering the concepts introduced in this chapter, including types of matrices, their operations, and properties, students will be well-prepared to tackle complex problems not just in mathematics, but also in fields such as engineering, computer science, and economics.

The resources provided by iPrep for Chapter 3 – Matrices, such as animated videos and practice questions, further enhance the learning experience, making it easier to grasp challenging concepts and apply them effectively. As you delve deeper into the world of matrices, remember that this foundational knowledge will serve as a stepping stone to even more advanced studies in mathematics and related disciplines. Embrace the learning journey in Chapter 3 – Matrices, and watch your mathematical abilities flourish!