Matrices

Understanding Matrix Multiplication and Other Operations

An array of numbers arranged in rectangular fashion and divided between rows and columns is called a matrix in mathematics. They are usually represented by writing all the numbers contained in it within square braces. There are many types of matrices and many operations like matrix multiplication which serve as crucial topics for boards and other entrance exams.

This is one of the most vital chapters in your maths syllabus. Almost all branches of studies which derive elements from mathematics, especially computer science, use this same concept thoroughly. For example, the figure below is that of a matrix with ‘m’ horizontal rows and ‘n’ vertical columns. 

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Figure 1: Example of a matrix


Different Types of Matrices

  1. Column Matrix – A matrix which has elements only in one column is called a column matrix. 

\[\begin{bmatrix} 1\\ 0\\ -5 \end{bmatrix}\]

Figure 2: Column Matrix

  1. Row Matrix – A matrix which has elements only in one row is called a row matrix. 

\[\begin{bmatrix} 1 & 5 & 9 \end{bmatrix}\]

Figure 3: Row Matrix

  1. Invertible Matrix – A matrix A of size b x b is called an invertible matrix only when another matrix B exists of same size such that AB = BA = I, where I is the identity matrix (containing only 1s in the principal diagonal) of the same dimension. In such a scenario, B is termed as the inverse matrix of A and also represented as A-1.

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Figure 4: Invertible Matrix

  1. Singular Matrices – A matrix which has no inverse (from the previous definition) is called singular matrix. Determinant value of singular matrix is always 0. For example, the below matrix is singular because its determinant = 0.

For example:

\[\begin{pmatrix} 3 &12 \\ 2 & 8 \end{pmatrix}\]

The determinant is = (3 x 8) - (12 x 2)

= 24 - 24

= 0

Figure 5: Singular Matrix

  1. Symmetric and Skew Symmetric Matrix – A matrix is called symmetric matrix if xij = xji, for all i and j, where xij = Element at ith row and jth column. Alternatively, a matrix is also called a symmetric matrix when its transpose is equal to the original matrix, AT=A. For example, the below matrix is symmetric because of the above conditions.

A = \[\begin{bmatrix} 3 & -2 & 4\\ -2 & 6& 2\\ 4& 2 & 3 \end{bmatrix}\]

Figure 6: Symmetric Matrix

Skew symmetric matrix is a matrix which satisfies the condition, AT= -A. 


Pop Quiz 1

  1. A matrix is a _______ array of numbers. 

    1. Rectangular (Answer)

    2. Square

    3. Circular

    4. None of the above


  1. What is the most unique property of skew symmetric matrices?

    1. AT= A

    2. AT= -A (Answer)

    3. AT + A = I

    4. AT. A = 0


Matrix Multiplication with a Scalar Number

A matrix can be multiplied with scalar numbers. If A = [aij]mxn (a matrix of size mxn) and k is a scalar which is to be multiplied to A, then the resultant matrix is obtained when each of the elements of A is multiplied with k, such that kA = [kaij]mxn. For example, take a look at the figure below. 

e.g    k\[\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}\]2x2  = \[\begin{bmatrix} ka_{11} & ka_{12}\\ ka_{21} & ka_{22} \end{bmatrix}\]2x2


Matrix Multiplication Between Two Matrices

If A = [aij]m x n and B = [bij]n x p are two matrices such that number of columns of A = number of rows of B, then the product of A and B is Cm x p. Each element cij of C is calculated with the formula below.  

\[C_{ij}\] = \[\sum_{h=1}^{n}a_{ik}b_{kj}\]


Properties for Multiplying Matrices

  1. Multiplying two matrices can only happen when the number of columns of the first matrix = number of rows of second matrix and the dimension of the product, hence, becomes (no. of rows of first matrix x no. of columns of second matrix). 

  2. In matrix multiplication, order must be maintained as said in point #1. Without this order, multiplication cannot take place. 

  3. In matrix multiplication, associative rule states that (AB)C = A(BC).

  4. In matrix multiplication, commutative rule states that AB ≠ BA. 

Exercise:

Take the following example and compute BC and A.(BC).

A = \[\begin{bmatrix} 1 & 0\\ 2 & 3\\ 3 & 1 \end{bmatrix}\]  B = \[\begin{bmatrix} 1 &2 & 1& 0\\ 0 & 1 & 0 & 2 \end{bmatrix}\]   C = \[\begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}\]

So, this was all about matrices and all other operations and important types of them, which will be needed for your exams. To know more about other topics of mathematics, visit the Vedantu website or download the app today. We host such easy-to-read tutorials and other important guides there.

FAQ (Frequently Asked Questions)

1. What Is A Matrix And Why Is It Such An Important Concept In Mathematics?

A matrix is a collection of numbers, which are organised in rows and columns. In all matrices, rows and columns are not necessarily present as there are some types of them which do not follow the same rule. The dimension of a matrix is represented as (no. of rows x no. of columns). If A is a matrix with m rows and n columns, then it is symbolised as Am x n. The concept of matrices is essential both in mathematics and computer science.

2. What Are The Different Types Of A Matrix?

A column matrix is a matrix which has numbers as elements only in a single column, and a row matrix is a matrix which has numbers only in a single row. A square matrix is a matrix with an equal number of rows and columns. In contrast, a diagonal matrix is a special type of square matrix which has elements only in its leading diagonal, and all other elements are 0s. A scalar matrix is another special type of diagonal matrix in which all the diagonal elements are equal.

3. What Is An Invertible And A Singular Matrix?

A matrix A of size b x b is called an invertible matrix only when another matrix B exists of same size such that AB = BA = I, where I is the identity matrix (containing only 1s in the principal diagonal) of the same dimension. In such a scenario, B is termed as the inverse matrix of A and also represented as A-1.

A matrix which has no inverse (from the previous definition) is called a singular matrix. Determinant value of a singular matrix is always 0.

4. What Are The Most Important Properties Of Matrix Multiplication?

Matrix multiplication can only happen when the number of columns of the first matrix is equal to the number of rows of second matrix and the dimension of the product. This comes to be (no. of rows of first matrix x no. of columns of second matrix). This ordering of matrices should always be maintained; otherwise, two matrices cannot be multiplied according to formula. In matrix multiplication, associative rule states that (AB)C = A(BC) and commutative rule states that AB ≠ BA. In product AB, A is called pre-factor while B is called post-factor.