Matrix Multiplication: Definition, Types, Properties and Formula

A matrix is nothing but a rectangular table (which we generally call an Array) of arranged numbers, symbols, expressions in the form of rows and columns. Numerous operations can be conducted on matrices such as addition, subtraction, multiplication, etc. Multiplication of matrix additionally called Matrix Multiplication is one of those operations in linear algebra which are mainly performed on matrices. But every operation on the matrix comes with certain terms and conditions. Likewise, for Matrix Multiplication to be successful, matrices involved Let’s say A and B are the defined matrices, then both A and B should be compatible. Because at least 2 matrices are required to perform the operation of matrix multiplication, Hence Matrix Multiplication is a binary operation as well.

Read More: Statistics Formula

Key Terms:- Matrix multiplication, Expressions, Symbols, Numbers, Cross product, Matrices, Addition, Subtraction, Multiplication, Division, Numerous operations, Linear algebra

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Matrix Multiplication

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If A and B are the two matrices, then their product (X) is denoted by:

X = AB

Hence we can conclude that the multiplication of two matrices is the dot product of two matrices.

The video below explains this:

Matrices Detailed Video Explanation:

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Types of Matrices

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• Column Matrix: Column matrices are those in which any number of rows and only one column is present.
• Row Matrix: A row matrix contains any number of columns but only one row. 
• Horizontal Matrix: Horizontal matrix is one in which the number of rows is less than the number of columns.
• Vertical Matrix: It is a type of matrix where the number of rows is greater than the number of columns.
• Square Matrix: A square matrix is a matrix of an order ab, with condition satisfying, a=b.
• Rectangular Matrix: A rectangular matrix is a matrix having an order of cd, such that c d.
• Null/Zero Matrix: Null or zero matrix is a matrix of any order but all of its elements would be zero.
• Scalar Matrix: A scalar matrix is a square matrix in which the diagonal elements are equal while every non-diagonal element is zero.
• Diagonal Matrix: A diagonal matrix is a square matrix where P=a| || |a| || || |x| |, provided all the elements except the elements in the leading diagonals are zero.
• Principal Diagonal of a Matrix: The first diagonal element of the first row to the last diagonal element of the last row is known as the principal diagonal of a matrix in a square matrix. 
• Identity/Unit Matrix: Identity matrix also refers to a square matrix in which every diagonal element is equal to 1 and every non-diagonal element is 0.
• Singular Matrix: A singular matrix is a square matrix P when the determinant of P is denoted by det (P) or |P| is zero which means, |P|= 0 if otherwise, it is known as a non-singular matrix.
• Equal Matrices: Two matrices suppose, A and B are said to be equal if both have the same order and corresponding elements of the matrices are all equal.

Read More: Area of Parallelogram

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Concept of Matrix Compatibility

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Let’s say we have two matrices A and B, so they will be compatible only when the number of columns in A should be equal to the number of rows in B. Also if A is a matrix of order m×n and B is the matrix of order n×p, then the matrices A and B are compatible.

Matrix Multiplication Formula and Rules 

A . B

a×b b×c

While multiplying the matrices, the first row will be multiplied and then the successive rows will be filled accordingly. Consider two matrices of order 3×3,

A = [a b c d e f g h i]  and B = [j k l m n o p q r]

Then the resultant matrix multiplication would be:

AB = [a b c d e f g h i][j k l m n o p q r]

= [aj+bm+cp] [ak+bn+cq] [al+bo+cr] [dj+em+fp] [dk+en+fq] [dl+eo+fr] [gj+hm+ip] [gk+mn+iq] [gl+ho+ir]

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Matrix Multiplication Categorization

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• 2×2 Matrix Multiplication

A = a₁ b₁ c₁ d₁ and B = a₂ b₂ c₂ d₂

AB = (a₁a₂+b₁c₂) (a₁b₂ + b₁d₂) (c₁a₂+d₁c₂) (c₁b₂+d₁d₂)

• 3×3 Matrix Multiplication

A = [a₁ b₁ c₁ d₁ e₁ f₁ g₁ h₁ i₁] and B = [a₂ b₂ c₂ d₂ e₂ f₂ g₂ h₂ i₂]

Then AB= [(a₁a₂+b₁d₂+c₁g₂) (a₁b₂+b₁e₂+c₁h₂) (a₁c₂+b₁f₂+c₁i₂) (d₁a₂+e₁d₂+f₁g₂) (d₁b₂+e₁e₂+f₁h₂) (d₁c₂+e₁f₂+f₁i₂) (g₁a₂+h₁d₂+i₁g₂) (g₁b₂+h₁e₂+i₁h₂) (g₁c₂+h₁f₂+i₁i₂)]

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Properties of Matrix Multiplication

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Matrix multiplication comes with quite a wide variety of properties, some of which are below.

• Commutative Property

The multiplication of matrices is non=commutative in nature. Let’s say there are two matrices namely A and B. Then AB ≠ BA.

Let’s look at one example to clearly understand this.

A = [1 2 3 4] and B = [5 6 7 8]

AB = [1 2 3 4] [5 6 7 8] =>[1×5+2×7 1×6+2×8 3×5+4×7 3×6+4×8] =>[14 22 43 50]

BA = [1 2 3 4] [5 6 7 8] =>[5×1+6×3 5×2+6×4 7×1+8×3 7×2+8×4]

=[23 34 31 46]

Hence it’s clearly proved that AB ≠ BA

• Associative Property

The multiplication of matrices is associative in nature i.e. (AB) C = A (BC)

Let’s look at one example to clearly understand this.

A = [1 2 3 4] , B = [5 6 7 8] , C = [9 10 11 12]

AB = [1 2 3 4] [5 6 7 8] = [19 22 43 50] and

BC = [5 6 7 8] [9 10 11 12] => [111 122 151 166]

Now (AB) C = [19 22 43 50] [9 10 11 12] => [171+242 190+264 387+550 430+600]

=> [413 454 937 1030]

A(BC) = [1 2 3 4] [111 122 151 166] => [111+302 122+332 333+604 366+664]

=> [413 454 937 1030]

Clearly, (AB) C = A (BC)

• Distributive Property

Matrix Multiplication is distributive in nature i.e. A (B+C) = AB + AC

A = [1 2 3 4] , B = [5 3 2 4] , C = [1 3 5 7]

B+C = [5 3 2 4 + 1 3 5 7] => [6 6 7 11]

A(B+C) = [1 2 3 4] [6 6 7 11] => [20 28 46 62]

AB = [1 2 3 4] [5 3 2 4] => [9 11 23 25]

AC = [1 2 3 4] [1 3 5 7] => [11 17 23 37]

AB + BC = [9 11 23 25] + [11 17 23 37] => [20 28 46 62]

Clearly, A (B+C) = AB + AC

• Multiplicative Identity Property

Multiplicative identity property is, 1.I = I.A = A

Where I is an identity matrix of order n and A is a matrix of order m×n

A = [2 3 1 6] and I = [1 0 0 1] => [2 3 1 6] = A

• Matrix Multiplication by Scalar/Integer/Constant

When we multiply an integer with a matrix, the resultant is simply known as a Scalar Multiplication.

Let’s say A = [34 -109 5] is being multiplied by 4

4× A = [4 × 34 -109 5] => [1216 – 403 620]

• Transpose Property

Suppose we have two matrices A and B, then the transpose property is defined as below.

(AB)^T = B^T A^T, where T is the transpose

A = [1 2 3 4] and B = [1 3 5 7]

AB = [1 2 3 4] [1 3 5 7] => [1 + 103 + 143 + 209 + 28]

= [11 17 23 37]

(AB)^T = [11 23 17 37] , A^T = [1 3 2 4] and B^T = [1 5 3 7]

Therefore, B^T A^T = [1 5 3 7] [1 3 2 4] => [1 + 103 + 203 + 149 + 28]

= [11 23 17 37]

• Dimension Property

Suppose that Matrix A is of order m×n and matrix B is of order n×a

Then multiplication of A and B would be m×a

If A is of order 2×3 and B is of order 3×4, then multiplication of AB would be of order 2×4.

• Multiplication of Matrix by zero

Suppose, A = [11 23 17 37] and we have a zero matrix in which all the numbers would be zero i.e. 

[0 0 0 0]

Then the product would be equal to [0 0 0 0] with no doubt.

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Matrix Multiplication Algorithms

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There are mainly 4 kinds of algorithms as per the significant amount of research in the field of matrix multiplication.

• Iterative Algorithm
• Sub-Cubic Algorithm
• Divide and Conquer Algorithm
• Parallel and Distributed Algorithm

And for online multiplication, these are used in a wide variety of languages such as C, Java, etc.

Now, this is the time we must be knowing where these matrices are used in our real life.

Read More: Differential Equations Applications

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Applications of Matrices

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• Cryptography

The process of encrypting data so that only authorized and destined users can access and know the meaning of the message. Two keys are used in this process – private and public key, and hence comes the role of matrices.

• Wireless Communication

Matrices are used to define, extract/fetch and process the encoded form of data in the signal. We can easily predict and detect issues in wireless communication using matrices. Matrices can also be used to process and display digitally formed images.

• Computer Graphics

To project 3-dimensional images into 2-dimensional planes, metrics are used. Also, a digital image is nothing but the representation of various patterns combined and arranged in a defined manner in graphics.

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Things to Remember

• The first and foremost condition for successful matrix multiplication is that matrices should be compatible.
• If A is a matrix having order m×n while Matrix B is of order n×q, then their product would be equal to m×p.
• Rows by column multiplication are performed in the Matrix Multiplication process.
• If the resultant matrix of multiplication of any 2 given matrices is a zero matrix, then it is not compulsory to be a zero matrix for one of them.
• Matrix Multiplication is both associative and distributive.