What are Matrices?
A matrix is a rectangular array of numbers, variables, symbols, or expressions used for operations such as
subtraction, addition, and multiplication. A matrix's size (also known as its order) is determined by the number of
rows and columns. A matrix with 6 rows and 4 columns has the order 6 × 4 and is read as 6 by 4. For example, the
given matrix B is a 3 × 4 matrix, represented as [B]3×4:
Certain rules must be followed when executing certain matrix operations, such as adding or subtracting
only if they have the same number of rows and columns, and multiplying only if the columns in the first and rows
in the second are identical. Let us look at the many forms of matrices and their laws in depth.Matrices in Math
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,
'aij', 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:
Calculate Matrices
Matrices can be solved by executing operations such as addition, subtraction, and multiplication. The number
of rows and columns determines how matrices are calculated. For addition and subtraction, the number of rows and
columns must be the same; for multiplication, the number of columns in the first matrix and the number of rows in
the second matrix must be identical. The fundamental operations that can be executed on matrices are:
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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) = [aij] + [bij] = [aij + bij], where i and j
are the number of rows and columns respectively. For example:
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) = [aij] - [bij] = [aij - bij],
where i and j are the row number and column number respectively. For example:
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(Aij)
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,
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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=[r1 r2...rn] and a column vector C=
. Then the product of R and C can be
defined as
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:
To obtain the element a11 of AB, we multiply R1 of A with C1 of B:
To obtain the element a12 of AB, we multiply R1 of A with C2 of B:
To obtain the element a21 of AB, we multiply R2 of A with C1 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 aij in AB, we
multiply Ri in A with Cj in B:
Properties of Matrix Multiplication
There are different properties associated with the multiplication of matrices. For any three matrices A, B, and C:
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AB ≠ BA
A(BC) = (AB)C
A(B + C) = AB + AC
(A + B)C = AC + BC
AIm = A = AIn, for identity matrices Im and In.
Am×n×On×p× = Om×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,
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(AT)T = A
(A + B)T = AT + BT, A and B being of the same order.
(KA)T= KAT, K is any scalar(real or complex).
(AB)T= BTAT, 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,
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tr(AB) = tr(BA)
tr(A) = tr(AT)
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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
Then determinant formula of matrix A:
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 aij��� is denoted as Mij���. For example, for the given matrix,
minor of a12 of the matrix A=
is:
Similarly, we can find all the minors of the matrix and will get a minor matrix M of the given matrix A as:
Cofactor of Matrix
Cofactor of an element in the matrix A is obtained when the minor Mij of the matrix is multiplied with (-1)i+j. The
cofactor of a matrix is denoted as Cij. If the minor of a matrix is Mij, then the cofactor of the matrix would be:
On finding all the cofactors of the matrix, we will get a cofactor matrix C of the given matrix A:
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=
Then the minor matrix M of the given matrix would be:
We will get the cofactor matrix C of the given matrix A as:
Then the transpose of the cofactor matrix will give the adjoint of the given matrix:
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:
Where:
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|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=
is
calculated by: A-1 =
Let us find the inverse of the 3 × 3 matrix we have used in the previous
And on calculating the determinant, we will get |A| = -33
Types of Matrices
There are various types of matrices based on the number of elements and the arrangement of elements in them.
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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=
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Rectangular Matrix: A matrix having unequal number of rows and columns is called a rectangular matrix.
For example: B=
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Diagonal matrices: A matrix with all non-diagonal elements to be zeros is known as a diagonal matrix.
Example: A=
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Identity matrices: A diagonal matrix having all the diagonal elements equal to 1 is called an identity
matrix.
Example: B=
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Symmetric and skew-symmetric matrices:
Symmetric matrices: A square matrix D of size n×n is considered to be symmetric if and only if DT= D.
For example, D =
is a symmetric matrix because
Skew-symmetric matrices-A square matrix F of size n×n is considered to be skew-symmetric if and only if
FT= - F.
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Invertible Matrix: Any square matrix A is called invertible matrix, if there exists another matrix B, such
that, AB = BA = In, where In 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., AT = 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:
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=(A−1)B=(−1). To find the determinant of matrix A, we will follow the below steps:
Hence, |A| = 3 - 2 = 1
∵ |A| ≠0, it is possible to find the inverse of matrix A.
Now, by using the formula for finding the inverse of 2x2 matrix
Now to find the matrix X, we'll multiply A−1 and B. We get,
Hence, the value of matrix X is,
Matrices Formulas
There are different formulas associated with matrix operations depending upon the type of matrix. Some of
the matrices formulas are listed below:
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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 (Am) = (adj A)m,
adj (kA) = kn-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
EXAMPLE: USING MATRICES IN REAL-WORLD PROBLEMS
We have the table below, representing the equipment needs of two soccer teams.
We are also given the prices of the equipment, as shown in the table below.
We will convert the data to matrices. Thus, the equipment need matrix is written as
The cost matrix is written as
The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is
$3,840.
HOW TO: GIVEN A MATRIX OPERATION, EVALUATE USING A CALCULATOR
1. Save each matrix as a matrix variable
[A],[B],[C],..
2. Enter the operation into the calculator, calling up each matrix variable as needed.
3. If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will
display an error message.
EXAMPLE: USING A CALCULATOR TO PERFORM MATRIX OPERATIONS
Find AB−C given
On the matrix page of the calculator, we enter matrix A above as the matrix variable [A], matrix B above as the
matrix variable [B], and matrix C above as the matrix variable [C].
On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.
[A]×[B]−[C]
The calculator gives us the following matrix.
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Matrices - Solve, types, meaning, Examples | Matrix Definition. (n.d.).
Cuemath.https://www.cuemath.com/algebra/solve-matrices/
Admin. (2023, April 24). Matrix operations - addition, subtraction, multiplication. BYJUS.
https://byjus.com/jee/matrix-operations/
Lumen Learning. (n.d.). Matrices and Matrix Operations | College Algebra.
https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-matrices-and-matrixoperations/