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. Matrices can have any number of columns and rows. Different operations can be performed on matrices matrix addition, scalar multiplication, matrix multiplication, transposition, etc.
There are certain rules to be followed while performing these matrix operations, like matrices 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 the rectangular table which contains various numbers of rows and columns. Matrices are rectangularshaped arrays, for which different operations like addition, multiplication, transposition are defined. The numbers or entries in the matrix are known as its elements. Horizontal entries for matrices are called rows and vertical entries are known as columns.
Definition of Matrix
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 is the number of rows and columns in the matrix. A matrix with 6 rows and 4 columns is represented as a 6 × 4 matrix or 6 by 4 matrix where 6 × 4 is the dimension of the matrix. For example, the given matrix B is a 3 × 4 matrix and is written as \([{B}]_{3 \times 4}\):
\(B = \left[\begin{array}{ccc} 2 & 1 & 3 & 5 \\ 0 & 5 & 2 & 7\\ 1 & 1 & 2 & 9 \end{array}\right]\)
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:
\(A = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} .. .& a_{1n} \\ a_{21} & a_{22} & a_{23} ... & a_{2n} \\ a_{31} & a_{32} & a_{33} ...& a_{3n} \\ : & : & : & : \\ a_{m1} & a_{m2} & a_{m3} ...& a_{mn} \end{array}\right] \)
How to Solve Matrices?
We can solve matrices by performing operations on them like addition, subtraction, multiplication, and so on.
A general matrix may look like:
\[\begin{bmatrix}
a_{11}&a_{12}&\cdots &a_{1n} \\
a_{21}&a_{22}&\cdots &a_{2n} \\
\vdots & \vdots & \ddots & \vdots\\
a_{m1}&a_{m2}&\cdots &a_{mn}
\end{bmatrix}\]
For example,
\( \begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
\end{bmatrix}\)
We will learn about these operations in detail in the following sections.
How to Solve System of Equations Using Matrices?
We have two matrices A and B where A is known as the coefficient matrix and B is known as the constant matrix. There is a third matrix X containing all the variables of the equations; this matrix 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.
The matrix A is:
\(A = \begin{bmatrix}
1 & 1\\
2 & 3\\
\end{bmatrix}\)
The matrix X is:
\(X = \begin{bmatrix}
x\\
y\\
\end{bmatrix}\)
The matrix B is:
\(B = \begin{bmatrix}
8\\
10\\
\end{bmatrix}\)
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 \( X = (A^{1})B\). To find the inverse of A, we will need the determinant and adjoint of matrix A. To find the determinant of matrix A, we will follow the below steps:
\( A = \begin{vmatrix}
1 & 1\\
2 & 3\\
\end{vmatrix}\)
Hence, A = 3  2 = 1
\(\because\) \(A \neq 0\), it is possible to find the inverse of matrix A.
The adjoint of matrix A is found by finding each element in it.
Each element in the cofactor matrix is called a minor and found by taking the determinant of the elements leaving the row and column for which the number is to found. The matrix obtained is a cofactor matrix. To find the adjoint of matrix, we take the transpose of the cofactor matrix.
Hence, the adjoint of matrix is given as
\(Adj A = C^T\)
\(C = \begin{bmatrix}
C_{11} & C_{12}\\
C_{21} & C_{22}\\
\end{bmatrix}\)
Each term in a cofactor matrix is obtained by \(C_{ij} = (1)^{i+j} \times M_{ij}\), where \(i\) indicates the row, \(j\) the column and \(M_{ij}\) being the minor of the matrix.
In this case, the adjoint of the matrix A is
\(Adj A = \begin{bmatrix}
3 & 1\\
2 & 1\\
\end{bmatrix}\)
To find the inverse of A, we divide the adjoint of A by the determinant of A.
Hence, \(A^{1} = \dfrac{Adj A}{A}\)
Applying the same to our example we get,
\(A^{1} = \begin{bmatrix}
3 & 1\\
2 & 1\\
\end{bmatrix}\)
Now to find the matrix X, we'll multiply \(A^{1}\) and B. We get,
\[\begin{bmatrix}
3 & 1 \\
2 & 1
\end{bmatrix}
%
\begin{bmatrix}
8 \\
10
\end{bmatrix}
\
=
\begin{bmatrix}
14 \\
6
\end{bmatrix}
\]
Hence, the value of matrix X is,
\(X = \begin{bmatrix}
14\\
6\\
\end{bmatrix}\)
Matrices Formulas
There are different formulas associated with matrix operations depending upon the type of matrix. Some of the matrices formulas are listed below:
i) A(adj A) = (adj A) A =  A  I\(_n\)
ii)  adj A  =  A ^{n1}
iii) adj (adj A) =  A ^{n2} A
iv)  adj (adj A)  =  A ^{(n1)^2}
v) adj (AB) = (adj B) (adj A)
vi) adj (A^{m}) = (adj A)^{m},
vii) adj (kA) = k^{n1} (adj A) , k ∈ R
viii) adj(I\(_n\)) = I\(_n\)
ix) adj 0 = 0
x) A is symmetric ⇒ (adj A) is also symmetric.
xi) A is diagonal ⇒ (adj A) is also diagonal.
xii) A is triangular ⇒ adj A is also triangular.
xiii) A is singular ⇒ adj A  = 0
ix) A^{1} = (1/A) adj A
x) (AB)^{1} = B^{1}A^{1}
Types of Matrices
There are various types of matrices based on the number of elements and the arrangement of elements in the matrices.
Row matrix: A matrix having a single row is called a row matrix. Example: [1, −2, 4].
Column matrix: 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 having only diagonal elements as nonzero numbers 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 skewsymmetric matrices:
Symmetric matrices: A square matrix D of size n×n is considered to be symmetric if and only if D^{T}= D. Let's consider the examples of two matrices D and F: D = \(\left[\begin{array}{lll} 2 & 3 & 6 \\ 3 & 4 & 5 \\ 6 & 5 & 9 \end{array}\right] \)
D^{T} = \(\left[\begin{array}{lll} 2 & 3 & 6 \\ 3 & 4 & 5 \\ 6 & 5 & 9 \end{array}\right]\)
Skewsymmetric matricesA square matrix F of size n×n is considered to be skewsymmetric if and only if F^{T}=  F.
\(F = \left[\begin{array}{ccc} 0 & 3 \\ \\ 3 & 0 \end{array}\right]\)
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,
A^{T} = A^{1}
Operations on Matrices
Any two matrices can be added, subtracted, and multiplied with each other depending 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 or matrices addition can only be possible if the number of rows and columns of both the matrices are the same. In adding two matrices, we add the elements in each row and column to the respective elements in the row and column of the next matrix. Hence (A + B) = [a\(_{ij}\)] + [b\(_{ij}\)] = [a\(_{ij}\) + b\(_{ij}\)], where i and j are the number of rows and columns respectively. For example: \(\begin{bmatrix} 2 & {1}\\ \\ 0 & 5\end{bmatrix} + \begin{bmatrix} 0 & 2 \\ \\ 1 & 2 \end{bmatrix}\\ = \begin{bmatrix} 2+0 & {1} +2 \\ \\ 0+1 & 5+(2) \end{bmatrix}\\ = \begin{bmatrix} 2 & 1 \\ \\1 & 3 \end{bmatrix} \)
Subtraction of Matrices
Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same. In subtracting two matrices, we subtract the elements in each row and column from the respective elements in the row and column of the previous matrix. Hence, (A  B) = [a\(_{ij}\)]  [b\(_{ij}\)] = [a\(_{ij}\)  b\(_{ij}\)], where i and j are the number of rows and columns respectively. For example: \( \begin{bmatrix} 2 & {1}\\ \\ 0 & 5 \end{bmatrix} \begin{bmatrix} 0 & 2 \\ \\1 & 2 \end{bmatrix} \\ = \begin{bmatrix} 20 & {1} 2\\ \\ 01 & 5(2) \end{bmatrix} \\ = \begin{bmatrix} 2 & 3\\ \\ 1 & 7 \end{bmatrix} \)
Scalar Multiplication
The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, 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
\(R=\left[ {{r}_{1}}\ {{r}_{2}}...{{r}_{n}} \right]\)
Note: A row vector is matrix with just one row.
and a column vector
\(C=\left[ \begin{align} \; \ {{c}_{1}} \;\\ \; \ {{c}_{2}} \; \\ \; \ \ \vdots \; \ \\ \; \ {{c}_{n}} \;\ \\ \end{align} \right]\)
Note: A column vector is a matrix with just one column.
which are both of order n. The product of R and C can be defined as
\(RC=\left[ {{r}_{1}}\ \ {{r}_{2}}\ \ ...\ {{r}_{n}} \right]\ \left[ \begin{align} & \ {{c}_{1}} \\ & \ {{c}_{2}} \\ & \ \ \vdots \ \\ & \ {{c}_{n}}\ \\ \end{align} \right]\ \\ =[{{r}_{1}}{{c}_{1}}+{{r}_{2}}{{c}_{2}}+...+{{r}_{n}}{{c}_{n}}]\)
Therefore, RC is a scalar quantity. For example,
\(\left[ 1\ \ 3\ \ 2 \right]\ \ \left[ \begin{align} & \ \ 2 \\ & 1 \\ & \ \ 4 \\ \end{align} \right]=[7]\)
Now, we will discuss matrix multiplication. It will soon become evident that to multiply two 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:\(A=\left[ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix} \right]B=\left[ \begin{matrix} {{\alpha }_{1}} & {{\beta }_{1}} \\ {{\alpha }_{1}} & {{\beta }_{2}} \\ {{\alpha }_{3}} & {{\beta }_{3}} \\\end{matrix} \right]\)
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. The important properties are listed below,
 AB ≠ BA, given matrices A and B.
 A(BC) = (AB)C, given matrices A, B and C.
 A(B + C) = AB + AC, for matrices A, B and C.
 (A + B)C = AC + BC, given matrices A, B and C.
 A\(I_m\) = A = AI\(_n\), given matrix A and identity matrices I\(_m\) and I\(_n\).
 A\(_{m\times n}\)O\(_{n\times p}\) = O\(_{m\times p}\), where A is a m × n matrix and 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 row1: 2, 3, 4, and row2: 1, 7, 7. On transposing, we will get the elements in column1: 2, 3, 4, and column2: 1, 7, 7, we can check that in the image given below:
Properties of transposition in matrices
There are various properties associated with the transposition operation in matrices, 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^{T}A^{T}, A and B being conformable for the product AB. (This is also called reversal law.)
Rank of a Matrix
The rank of any 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.
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) = ctr(A)
 tr(A + B) = tr(A) + tr(B)
Minor of Matrix
Minor of matrix for a particular element in the matrices is defined as the matrix 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:
\(M_{12} = \left\begin{array}{ccc} a_{21} & a_{23} \\ a_{31} & a_{33} \end{array}\right\)
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 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.
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:
\(a_{11}(1)^{1 + 1} \!\!\left\begin{matrix}a_{22}\!\!\!&a_{23}\\a_{32}\!\!\!&a_{33}\end{matrix}\right \!\!+\!\! a_{12}(1)^{1 + 2} \!\!\left\begin{matrix}a_{21}\!\!\!&a_{23}\\a_{31}\!\!\!&a_{33}\end{matrix}\right \!\!+\!\! a_{13}(1)^{1 + 3} \!\!\left\begin{matrix}a_{21}\!\!\!&a_{22}\\a_{31}\!\!\!&a_{32}\end{matrix}\right\)
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 or the inverse of matrices 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 \(A = \left[\begin{array}{ccc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] \) is calculated by: A^{1} = \(\dfrac{1}{a_{11}a_{22}  a_{12}a_{21}}\left(\begin{matrix}a_{22}&a_{12}\\a_{21}&a_{11}\end{matrix}\right)\)
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} = \(\left[\begin{array}{ccc}
0.24 & 0.15 & 0.51 \\
0.06 & 0.21 & 0.12 \\
0.15 & 0.03 & 0.39
\end{array}\right] \)
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 eigen value and a nonzero vector v, called the eigen vector, satisfy the equation, Av = λv. λ is an eigenvalue of an n×nmatrix A if and only if A − λI\(_n\) is not invertible, which is equivalent to Det(A  λI) = 0.
Matrices Tips and Tricks:
 Cofactor of the matrix A is obtained when the minor \(M_{ij}\) of the matrix is multiplied with (1)^{i+j}.
 Matrices are rectangularshaped 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.
Related Topics:
Check out these interesting articles to know more about matrices:
Solved Examples on Matrices

Example 1: Let \(A=\left[ \begin{matrix} 1 & 2\\ \\ 3 & 1 \\\end{matrix} \right],\ B=\left[ \begin{matrix} 1 & 4\\ \\ 3 & 1 \\\end{matrix} \right]\)
Calculate A + B.
Solution:
Here, matrix A = \(\left[ \begin{matrix} 1 & 2\\ \\ 3 & 1 \\\end{matrix} \right]\)
matrix B = \(\left[ \begin{matrix} 1 & 4\\ \\ 3 & 1 \\\end{matrix} \right]\)Using addition of matrices property, A + B = \(\left[ \begin{matrix} 1 & 2 \\ \\ 3 & 1 \\\end{matrix} \right]\) + \(\left[ \begin{matrix} 1 & 4\\ \\ 3 & 1 \\\end{matrix} \right]\) = \(\left[ \begin{matrix} 2 & 6\\ \\ 6 & 0 \end{matrix} \right]\)
Answer: Sum of matrices A and B, A + B = \(\left[ \begin{matrix} 2 & 6\\ \\ 6 & 0 \end{matrix} \right]\)

Example 2: Find the inverse of a matrix A =\(\left[\begin{matrix}1 & 2\\ \\2 & 3 \end{matrix}\right]\).
Solution:
The given matrix is A = \(\left[\begin{matrix}1 & 2\\ \\2 & 3 \end{matrix}\right]\).
Using the formula of matrix inverse: A^{1} = \(\dfrac{1}{a_{11}a_{22}  a_{12}a_{21}}\left[\begin{matrix}a_{22}&a_{12}\\ \\a_{21}&a_{11}\end{matrix}\right]\)
Using the inverse of matrix formula we can calculate A^{1} as follows.
A^{1} = \(\dfrac{1}{(1× 3)  (2 × 2)}\left[\begin{matrix}3&2\\ \\2&1\end{matrix}\right]\)
= \(\dfrac{1}{3 +4}\left[\begin{matrix}3&2\\ \\2&1\end{matrix}\right]\)
= \(\left[\begin{matrix}3&2\\ \\2&1\end{matrix}\right]\)
Answer: Therefore A^{1} = \(\left[\begin{matrix}3&2\\ \\2&1\end{matrix}\right]\).
FAQs on Matrices
What are Matrices in Math?
Matrices in math are arrangements of numbers, variables, symbols, or expressions in the rectangular table which contains various numbers of rows and columns, for which the operations like addition, multiplication, transposition, etc are defined.
How to Solve Matrices?
We can solve matrices by performing operations on them like addition, subtraction, multiplication, and so on.
How to Solve Systems of Equations with Matrices?
To solve the system of equations with matrices, we will follow the steps given below.
 Arrange the elements of equations in matrices and find the coefficient matrix, variable matrix, and constant matrix.
 Write the equations in AX = B form.
 Take the inverse of A by finding the adjoint and determinant of A.
 Multiply the inverse of A to matrix B, thereby finding the value of variable matrix X.
What is 3×3 Inverse Matrix Formula?
The inverse matrix formula for a 3×3 matrix is, A^{1} = adj(A)/A; A ≠ 0 where A = square matrix, adj(A) = adjoint of square matrix, A^{1} = inverse matrix.
What is the Special Feature Of the Determinant Formula For Matrices?
The determinant of a matrix is defined only for square matrices, and this property of the determinant formula makes it unique. Also, the determinant value can be calculated by using the elements of any row or any column.
How To Calculate the Determinant of a 2×2 Matrix Using Determinant Formula?
To calculate the determinant of a 2×2 matrix
 Step 1: Check if the given matrix is a square matrix that too a 2×2 matrix.
 Step 2: Identify all its rows and columns.
 Step 3: Put the values in the determinant formula, D = ad  bc.
The determinant formula for 2 x 2 matrix, \(A =\begin{pmatrix}a &b\\ \\c&d\end{pmatrix}\) is given by D = ad  bc.
What is the Condition for Matrix Multiplication to be Possible?
Matrix multiplication is possible only if the matrices are compatible i.e., matrix multiplication is valid only if the number of columns of the first matrix is equal to the number of rows of the second matrix.
What Are Properties of Transposition of Matrices?
For given two matrices, A and B, the properties of transposition of matrices can be explained as given below,
 (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^{T}A^{T}, A and B being conformable for the product AB. (This is also called reversal law.)
What is the Formula for Inverse of Matrices?
The inverse matrix formula is used to determine the inverse matrix for any given matrix. The inverse of a square matrix, A is A^{1}. The inverse matrix formula can be given as, A^{1} = adj(A)/A; A ≠ 0, where A is a square matrix. Also for a matrix and its inverse we have A × A^{1} = A^{1} × A = I.
How To Use Inverse of Matrix Formula?
The inverse matrix formula can be used following the given steps:
 Step 1: Find the matrix of minors for the given matrix.
 Step 2: Transform the minor matrix so obtained into the matrix of cofactors.
 Step 3: Find the adjoint matrix by taking the transpose of the cofactor matrix.
 Step 4: Finally divide the adjoint of a matrix with its determinant.
What are the Different Types of a Matrix?
There are different types of matrices depending upon the properties of their properties. Some of them are given as,
 Row matrix and Column matrix
 Square matrix and Rectangular matrix
 Diagonal Matrix
 Scalar Matrix
 Identity matrix
 Null matrix
 Upper triangular matrix and lower triangular matrix
 Idempotent matrix
 Symmetric and Skewsymmetric matrix
What are the Properties of Scalar Multiplication in Matrices?
Given the matrices A = [a\(_{ij}\)]\(_{m\times n}\) and B = [b\(_{ij}\)]\(_{m\times n}\) and scalars K and l, the different properties associated with the multiplication of matrices can be 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
What is a Matrix Polynomial?
Given a polynomial of the form, f(x) = a\(_0\)x^{n} + a\(_1\)x^{n1} + a\(_2\)x^{n1}+ . . . + a\(_{n1}x\) + a\(_n\), and A as a square matrix of order n. Then, f(A) = a\(_0\)A^{n} + a\(_1\)A^{n1} + a\(_2\)A^{n2} + . . . + a\(_{n1}\)A + a\(_n\)A + a\(_n\)I\(_n\) is called the matrix polynomial.
What is the Echelon Form of Matrices?
A matrix A = (a\(_{ij}\)\(_{m\times n}\) is said to be of echelon form, if all the nonzero rows, if any, precede the zero rows and the number of zeros preceding the first nonzero element in a row is less than the number of such zeros in the succeeding row.
How to Express a Matrix as a Sum of Symmetric and NonSymmetric Matrix?
Any square matrix A can be written as, A = P + Q, where P and Q are symmetric and skewsymmetric matrices respectively, such that, P = (A + A^{T})/2 and Q = (A  A^{T})/2.
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