Matrices

Matrices is a plural form of a matrix, which is a rectangular array or a table where numbers or elements are arranged in rows and columns. They can have any number of columns and rows. Different operations can be performed on matrices such as addition, scalar multiplication, multiplication, transposition, etc.

There are certain rules to be followed while performing these matrix operations like they can be added or subtracted if only they have the same number of rows and columns whereas they can be multiplied if only columns in first and rows in second are exactly the same. Let us understand the different types of matrices and these rules in detail.

What are Matrices?

Matrices, the plural form of a matrix, are the arrangements of numbers, variables, symbols, or expressions in a rectangular table that contains various numbers of rows and columns. They are rectangular-shaped arrays, for which different operations like addition, multiplication, and transposition are defined. The numbers or entries in the matrix are known as its elements. Horizontal entries of matrices are called rows and vertical entries are known as columns.

Matrix Definition

A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix (which is known as the order of the matrix) is determined by the number of rows and columns in the matrix. The order of a matrix with 6 rows and 4 columns is represented as a 6 × 4 and is read as 6 by 4. For example, the given matrix B is a 3 × 4 matrix and is written as $[B{]}_{3\times 4}$:

$B=\left[\begin{array}{cccc}2& -1& 3& 5\\ 0& 5& 2& 7\\ 1& -1& -2& 9\end{array}\right]$

Notation of Matrices

If a matrix has m rows and n columns, then it will have m × n elements. A matrix is represented by the uppercase letter, in this case, 'A', and the elements in the matrix are represented by the lower case letter and two subscripts representing the position of the element in the number of row and column in the same order, in this case, '$a_{ij}$', where i is the number of rows, and j is the number of columns. For example, in the given matrix A, element in the 3rd row and 2nd column would be $a_{32}$, can be verified in the matrix given below:

$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}...& a_{1n}\\ a_{21}& a_{22}& a_{23}...& a_{2n}\\ a_{31}& a_{32}& a_{33}...& a_{3n}\\ :& :& :& :\\ a_{m1}& a_{m2}& a_{m3}...& a_{mn}\end{array}\right]$

Calculate Matrices

We can solve matrices by performing operations on them like addition, subtraction, multiplication, and so on. Calculating matrices depends upon the number of rows and columns. For addition and subtraction, the number of rows and columns must be the same whereas, for multiplication, number of columns in the first and the number of rows in the second matrix must be equal. The basic operations that can be performed on matrices are:

• Addition of Matrices
• Subtraction of Matrices
• Scalar Multiplication
• Multiplication of Matrices
• Transpose of Matrices

Addition of Matrices

The addition of matrices can only be possible if the number of rows and columns of both the matrices are the same. While adding 2 matrices, we add the corresponding elments. i.e., (A + B) = [a${}_{ij}$] + [b${}_{ij}$] = [a${}_{ij}$ + b${}_{ij}$], where i and j are the number of rows and columns respectively. For example:$\left[\begin{array}{cc}2& -1\\ & \\ 0& 5\end{array}\right]+\left[\begin{array}{cc}0& 2\\ & \\ 1& -2\end{array}\right]=\left[\begin{array}{cc}2+0& -1+2\\ & \\ 0+1& 5+(-2)\end{array}\right]=\left[\begin{array}{cc}2& 1\\ & \\ 1& 3\end{array}\right]$

Subtraction of Matrices

Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same. While subtracting 2 matrices, we subtract the corresponding elements. i.e., (A - B) = [a${}_{ij}$] - [b${}_{ij}$] = [a${}_{ij}$ - b${}_{ij}$], where i and j are the row number and column number respectively. For example:$\left[\begin{array}{cc}2& -1\\ & \\ 0& 5\end{array}\right]-\left[\begin{array}{cc}0& 2\\ & \\ 1& -2\end{array}\right]=\left[\begin{array}{cc}2-0& -1-2\\ & \\ 0-1& 5-(-2)\end{array}\right]=\left[\begin{array}{cc}2& -3\\ & \\ -1& 7\end{array}\right]$

Scalar Multiplication

The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication. i.e., (cA)${}_{ij}$ = c(A${}_{ij}$)

Properties of scalar multiplication in matrices

The different properties of matrices for scalar multiplication of any scalars K and l, with matrices A and B are given as,

• K(A + B) = KA + KB
• (K + l)A = KA + lA
• (Kl)A = K(lA) = l(KA)
• (-K)A = -(KA) = K(-A)
• 1·A = A
• (-1)A = -A

Multiplication of Matrices

Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal. To understand how matrices are multiplied, let us first consider a row vector $R=[r_1{r}_2...r_n]$ and a column vector $C=\left[\begin{array}{c}{c}_1\\ c_2\\ ⋮\\ c_n\end{array}\right]$. Then the product of R and C can be defined as

$RC=[r_1{r}_2...r_n]\left[\begin{array}{cc}& c_1\\ & c_2\\ & ⋮\\ & c_n\end{array}\right]=[r_1{c}_1+r_2{c}_2+...+r_n{c}_n]$. For example,

$\left[132\right]\left[\begin{array}{cc}& 2\\ & -1\\ & 4\end{array}\right]=\left[7\right]$

Now, we will discuss matrix multiplication. It will soon become evident that to multiply 2 matrices A and B and to find AB, the number of columns in A should equal the number of rows in B.

Let A be of order m × n and B be of order n × p. The matrix AB will be of order m × p and will be obtained by multiplying each row vector of A successively with column vectors in B. Let us understand this using a concrete example:$A=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]B=\left[\begin{array}{cc}{\alpha }_1& {\beta }_1\\ {\alpha }_1& {\beta }_2\\ {\alpha }_3& {\beta }_3\end{array}\right]$

To obtain the element $a_{11}$ of AB, we multiply $R_1$ of A with $C_1$ of B :

To obtain the element $a_{12}$ of AB, we multiply $R_1$ of A with $C_2$ of B:

To obtain the element $a_{21}$ of AB, we multiply $R_2$ of A with $C_1$ of B:

Proceeding this way, we obtain all the elements of AB.

Let us generalize this: if A is or order m × n, and B of order n × p, then to obtain the element $a_{ij}$ in AB, we multiply $R_i$ in A with $C_j$ in B:

Properties of Matrix Multiplication

There are different properties associated with the multiplication of matrices. For any three matrices A, B, and C:

• AB ≠ BA
• A(BC) = (AB)C
• A(B + C) = AB + AC
• (A + B)C = AC + BC
• A$I_m$ = A = AIn, for identity matrices I${}_m$ and In.
• A${}_{m\times n}$O${}_{n\times p}$ = O${}_{m\times p}$, where O is a null matrix.

Transpose of Matrix

The transpose of a matrix is done when we replace the rows of a matrix to the columns and columns to the rows. Interchanging of rows and columns is known as the transpose of matrices. In the matrix given below, we have row elements as row-1: 2, -3, -4, and row-2: -1, 7, -7. On transposing, we will get the elements in column-1: 2, -3, -4, and column-2: -1, 7, -7, we can check that in the image given below:

Properties of transposition in matrices

There are various properties associated with transposition. For matrices A and B, given as,

• (A^T)^T = A
• (A + B)^T = A^T + B^T, A and B being of the same order.
• (KA)^T= KA^T, K is any scalar(real or complex).
• (AB)^T= B^TA^T, A and B being conformable for the product AB. (This is also called reversal law.)

Apart from these operations, we have several other operations on matrices like finding its trace, determinant, minors and cofactors, adjoint, inverse, etc. Let us learn each of these in detail in the upcoming sections.

Trace of a Matrix

The trace of any matrix A, Tr(A) is defined as the sum of its diagonal elements. Some properties of trace of matrices are,

• tr(AB) = tr(BA)
• tr(A) = tr(A^T)
• tr(cA) = c tr(A), for a scalar 'c'
• tr(A + B) = tr(A) + tr(B)

Determinant of Matrices

The determinant of a matrix is a number defined only for square matrices. It is used in the analysis of linear equations and their solution. The determinant formula helps calculate the determinant of a matrix using the elements of the matrix. Determinant of a matrix is equal to the summation of the product of the elements of a particular row or column with their respective cofactors. Determinant of a matrix A is denoted as |A|. Let say we want to find the determinant of the matrix $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$

Then determinant formula of matrix A:

$a_{11}(-1{)}^{1+1}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|+a_{12}(-1{)}^{1+2}\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|+a_{13}(-1{)}^{1+3}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$

Minor of Matrix

Minor for a particular element in the matrices is defined as the determinant of the matrix that is obtained when the row and column of the matrix in which that particular element lies are deleted, and the minor of the element $a_{ij}$ is denoted as $M_{ij}$. For example, for the given matrix, minor of $a_{12}$ of the matrix $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$ is:

$M_{12}=\left|\begin{array}{cc}{a}_{21}& a_{23}\\ & \\ a_{31}& a_{33}\end{array}\right|$

Similarly, we can find all the minors of the matrix and will get a minor matrix M of the given matrix A as:

$M=\left[\begin{array}{ccc}{M}_{11}& M_{12}& M_{13}\\ M_{21}& M_{22}& M_{23}\\ M_{31}& M_{32}& M_{33}\end{array}\right]$

Cofactor of Matrix

Cofactor of an element in the matrix A is obtained when the minor $M_{ij}$ of the matrix is multiplied with (-1)^i+j. The cofactor of a matrix is denoted as $C_{ij}$. If the minor of a matrix is $M_{ij}$, then the cofactor of the matrix would be:

$C_{ij}=(-1{)}^{i+j}{M}_{ij}$

On finding all the cofactors of the matrix, we will get a cofactor matrix C of the given matrix A:

$C=\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]$

Note: Be extra cautious about the negative sign while calculating the cofactor of the matrix.

Adjoint of Matrices

The adjoint of matrices is calculated by finding the transpose of the cofactors of the elements of the given matrices. To find the adjoint of a matrix, we have to calculate the cofactors of the elements of the matrix and then transpose the cofactor matrix to get the adjoint of the given matrix. The adjoint of matrix A is denoted by adj(A). Let us understand this with an example: We have a matrix $A=\left[\begin{array}{ccc}2& -1& 3\\ 0& 5& 2\\ 1& -1& -2\end{array}\right]$

Then the minor matrix M of the given matrix would be:

$M=\left[\begin{array}{ccc}-8& -2& -5\\ 5& -7& -1\\ -17& 4& 10\end{array}\right]$

We will get the cofactor matrix C of the given matrix A as:

$C=\left[\begin{array}{ccc}-8& 2& -5\\ -5& -7& 1\\ -17& -4& 10\end{array}\right]$

Then the transpose of the cofactor matrix will give the adjoint of the given matrix:

adj(A) = C^T = $\left[\begin{array}{ccc}-8& -5& -17\\ 2& -7& -4\\ -5& 1& 10\end{array}\right]$

Inverse of Matrices

The inverse of any matrix is denoted as the matrix raised to the power (-1), i.e. for any matrix "A", the inverse matrix is denoted as A^-1. The inverse of a square matrix, A is A^-1 only when: A × A^-1 = A^-1 × A = I. There is a possibility that sometimes the inverse of a matrix does not exist if the determinant of the matrix is equal to zero(|A| = 0). The inverse of a matrix is shown by A^-1. Matrices inverse is calculated by using the following formula:

A^-1 = (1/|A|)(Adj A)

where

• |A| is the determinant of the matrix A and |A| ≠ 0.
• Adj A is the adjoint of the given matrix A.

The inverse of a 2 × 2 matrix $A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ & \\ a_{21}& a_{22}\end{array}\right]$ is calculated by: A^-1 = $\frac{1}{a_{11}{a}_{22}-a_{12}{a}_{21}}\left(\begin{array}{cc}{a}_{22}& -a_{12}\\ & \\ -a_{21}& a_{11}\end{array}\right)$

Let us find the inverse of the 3 × 3 matrix we have used in the previous section: $A=\left[\begin{array}{ccc}2& -1& 3\\ 0& 5& 2\\ 1& -1& -2\end{array}\right]$

Since adj(A) = $\left[\begin{array}{ccc}-8& -5& -17\\ 2& -7& -4\\ -5& 1& 10\end{array}\right]$

And on calculating the determinant, we will get |A| = -33

Therefore, A^-1 = (1/-33) × $\left[\begin{array}{ccc}-8& -5& -17\\ 2& -7& -4\\ -5& 1& 10\end{array}\right]$

Hence, A^-1 = $\left[\begin{array}{ccc}0.24& 0.15& 0.51\\ -0.06& 0.21& 0.12\\ 0.15& -0.03& -0.39\end{array}\right]$

Types of Matrices

There are various types of matrices based on the number of elements and the arrangement of elements in them.

Row matrix: A row matrix is a matrix having a single row is called a row matrix. Example: [1, −2, 4].

Column matrix: A column matrix is a matrix having a single column is called a column matrix. Example: [−1, 2, 5]^T.

Square matrix: A matrix having equal number of rows and columns is called a square matrix. For example: $B=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 0& 6& 1\end{array}\right]$

Rectangular Matrix: A matrix having unequal number of rows and columns is called a rectangular matrix. For example: $B=\left[\begin{array}{ccc}1& 2& 3\\ & & \\ 0& 1& 4\end{array}\right]$

Diagonal matrices: A matrix with all non-diagonal elements to be zeros is known as a diagonal matrix.
Example: $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]$

Identity matrices: A diagonal matrix having all the diagonal elements equal to 1 is called an identity matrix.
Example: $B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Symmetric and skew-symmetric matrices:
Symmetric matrices: A square matrix D of size n×n is considered to be symmetric if and only if D^T= D. For example, D = $\left[\begin{array}{ccc}2& 3& 6\\ 3& 4& 5\\ 6& 5& 9\end{array}\right]$ is a symmetric matrix because

D^T = $\left[\begin{array}{ccc}2& 3& 6\\ 3& 4& 5\\ 6& 5& 9\end{array}\right]$ = D

Skew-symmetric matrices-A square matrix F of size n×n is considered to be skew-symmetric if and only if F^T= - F.

$F=\left[\begin{array}{cc}0& 3\\ & \\ -3& 0\end{array}\right]$ is a skew-symmetric matrix because

• F^T = $\left[\begin{array}{cc}0& -3\\ & \\ 3& 0\end{array}\right]$
• -F = $\left[\begin{array}{cc}0& -3\\ & \\ 3& 0\end{array}\right]$

Invertible Matrix: Any square matrix A is called invertible matrix, if there exists another matrix B, such that, AB = BA = $I_n$, where $I_n$ is an identity matrix with n × n.

Orthogonal Matrix: Any square matrix A is orthogonal if its transpose is equal to its inverse. i.e., A^T = A^-1

Solving a System of Equations Using Matrices

While solving the system of equations using matrices, we have three matrices A, B, and X where A is known as the coefficient matrix, B is known as the constant matrix, and X contains all the variables of the equations which is known as a variable matrix. Matrix A is of the order m × n, while B is the column matrix of the order m × 1. The product of matrix A and matrix X results in matrix B; hence, X is a column matrix as well of the order n × 1.

The matrices are arranged as:

A • X = B

Let's understand how to solve a system of equations using matrices with the help of an example. We have a set of two equations as given below. The equations are:

x + y = 8
2x + 3y = 10

Arrange all the coefficients, variables, and constants in the matrix in such a way that whenever we find the product of the matrices, the result obtained must result in the equation. Then the matrix equation is, AX = B where:

$A=\left[\begin{array}{cc}1& 1\\ & \\ 2& 3\end{array}\right]$

$X=\left[\begin{array}{c}x\\ \\ y\end{array}\right]$

$B=\left[\begin{array}{c}8\\ \\ 10\end{array}\right]$

To solve the equations, we need to find matrix X. It can be found by multiplying the inverse of matrix A with B, which is given as $X=\left(A^{-1}\right)B$. To find the determinant of matrix A, we will follow the below steps:

$|A|=\left|\begin{array}{cc}1& 1\\ & \\ 2& 3\end{array}\right|$

Hence, |A| = 3 - 2 = 1
$\because$ $|A|\ne 0$, it is possible to find the inverse of matrix A.

Now, by using the formula for finding the inverse of 2x2 matrix (which is mentioned in previous sections),

$A^{-1}=\left[\begin{array}{cc}3& -1\\ & \\ -2& 1\end{array}\right]$

Now to find the matrix X, we'll multiply $A^{-1}$ and B. We get,

$\left[\begin{array}{cc}3& -1\\ & \\ -2& 1\end{array}\right]\left[\begin{array}{c}8\\ \\ 10\end{array}\right]=\left[\begin{array}{c}14\\ \\ -6\end{array}\right]$

Hence, the value of matrix X is,

$X=\left[\begin{array}{c}14\\ \\ -6\end{array}\right]$

Rank of a Matrix

The rank of a matrix A is defined as the maximum number of linearly independent row(or column) vectors of the matrix. That means the rank of a matrix will always be less than or equal to the number of its rows or columns. The rank of a null matrix is zero since it has no independent row or column vectors.

Eigen Values and Eigen Vectors of Matrices

If A is any square matrix of order 'n', a matrix of A - λI can be formed, where I is a unit matrix of order n, such that the number λ, called the eigenvalue and a non-zero vector v, called the eigenvector, satisfy the equation, Av = λv. λ is an eigenvalue of an n×n-matrix A if and only if A − λIn is not invertible, which is equivalent to Det(A - λI) = 0.

Matrices Formulas

There are different formulas associated with matrix operations depending upon the type of matrix. Some of the matrices formulas are listed below:

• A(adj A) = (adj A) A = | A | In
• | adj A | = | A |^n-1
• adj (adj A) = | A |^n-2 A
• | adj (adj A) | = | A |^{(n-1)^2}
• adj (AB) = (adj B) (adj A)
• adj (A^m) = (adj A)^m,
• adj (kA) = k^n-1 (adj A) , k ∈ R
• adj(In) = In
• adj 0 = 0
• A is symmetric ⇒ (adj A) is also symmetric.
• A is diagonal ⇒ (adj A) is also diagonal.
• A is triangular ⇒ adj A is also triangular.
• A is singular ⇒| adj A | = 0
• A^-1 = (1/|A|) adj A
• (AB)^-1 = B^-1A^-1

Important Notes on Matrices:

• Cofactor of the matrix A is obtained when the minor $M_{ij}$ of the matrix is multiplied with (-1)^i+j.
• Matrices are rectangular-shaped arrays.
• The inverse of matrices is calculated by using the given formula: A^-1 = (1/|A|)(adj A).
• The inverse of a matrix exists if and only if |A| ≠ 0.