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.

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.

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 \([{B}]_{3 \times 4}\):

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

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:

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

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: \(\begin{bmatrix} 2 & {-1}\\ \\ 0 & 5\end{bmatrix} + \begin{bmatrix} 0 & 2 \\ \\ 1 & -2 \end{bmatrix}\\ = \begin{bmatrix} 2+0 & {-1} +2 \\ \\ 0+1 & 5+(-2) \end{bmatrix}\\ = \begin{bmatrix} 2 & 1 \\ \\1 & 3 \end{bmatrix} \)

Subtraction of Matrices

Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same. 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: \( \begin{bmatrix} 2 & {-1}\\ \\ 0 & 5 \end{bmatrix} -\begin{bmatrix} 0 & 2 \\ \\1 & -2 \end{bmatrix} \\ = \begin{bmatrix} 2-0 & {-1} -2\\ \\ 0-1 & 5-(-2) \end{bmatrix} \\ = \begin{bmatrix} 2 & -3\\ \\ -1 & 7 \end{bmatrix} \)

Scalar Multiplication

The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, 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 \(R=\left[ {{r}_{1}}\ {{r}_{2}}...{{r}_{n}} \right]\) and a column vector \(C=\left[ \begin{align} \; \ {{c}_{1}} \;\\ \; \ {{c}_{2}} \; \\ \; \ \ \vdots \; \ \\ \; \ {{c}_{n}} \;\ \\ \end{align} \right]\). Then the product of R and C can be defined as

\(RC=\left[ {{r}_{1}}\ \ {{r}_{2}}\ \ ...\ {{r}_{n}} \right]\ \left[ \begin{align} & \ {{c}_{1}} \\ & \ {{c}_{2}} \\ & \ \ \vdots \ \\ & \ {{c}_{n}}\ \\ \end{align} \right]\ \\ =[{{r}_{1}}{{c}_{1}}+{{r}_{2}}{{c}_{2}}+...+{{r}_{n}}{{c}_{n}}]\). 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 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:\(A=\left[ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix} \right]B=\left[ \begin{matrix} {{\alpha }_{1}} & {{\beta }_{1}} \\ {{\alpha }_{1}} & {{\beta }_{2}} \\ {{\alpha }_{3}} & {{\beta }_{3}} \\\end{matrix} \right]\)

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. 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 = AI_n, for identity matrices I\(_m\) and I_n.
• 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.

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)

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:

\(a_{11}(-1)^{1 + 1} \!\!\left|\begin{matrix}a_{22}\!\!\!&a_{23}\\a_{32}\!\!\!&a_{33}\end{matrix}\right| \!\!+\!\! a_{12}(-1)^{1 + 2} \!\!\left|\begin{matrix}a_{21}\!\!\!&a_{23}\\a_{31}\!\!\!&a_{33}\end{matrix}\right| \!\!+\!\! a_{13}(-1)^{1 + 3} \!\!\left|\begin{matrix}a_{21}\!\!\!&a_{22}\\a_{31}\!\!\!&a_{32}\end{matrix}\right|\)

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:

\(M_{12} = \left|\begin{array}{ccc} a_{21} & a_{23} \\ \\ a_{31} & a_{33} \end{array}\right|\)

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 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.

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

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 \(A = \left[\begin{array}{ccc} a_{11} & a_{12} \\ \\ a_{21} & a_{22} \end{array}\right] \) 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 = \(\left[\begin{array}{ccc}
0.24 & 0.15 & 0.51 \\
-0.06 & 0.21 & 0.12 \\
0.15 & -0.03 & -0.39
\end{array}\right] \)

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 = \(\left[\begin{array}{lll} 2 & 3 & 6 \\ 3 & 4 & 5 \\ 6 & 5 & 9 \end{array}\right] \) is a symmetric matrix because

D^T = \(\left[\begin{array}{lll} 2 & 3 & 6 \\ 3 & 4 & 5 \\ 6 & 5 & 9 \end{array}\right]\) = 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.

\(F = \left[\begin{array}{ccc} 0 & 3 \\ \\ -3 & 0 \end{array}\right]\) 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

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:

\(A = \begin{bmatrix}
1 & 1\\ \\
2 & 3\\
\end{bmatrix}\)

\(X = \begin{bmatrix}
x\\ \\
y\\
\end{bmatrix}\)

\(B = \begin{bmatrix}
8\\ \\
10\\
\end{bmatrix}\)

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 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.

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

\(A^{-1} = \begin{bmatrix}
3 & -1\\ \\
-2 & 1\\
\end{bmatrix}\)

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,

\(X = \begin{bmatrix}
14\\ \\
-6\\
\end{bmatrix}\)

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.

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 − λI_n is not invertible, which is equivalent to Det(A - λI) = 0.

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 | I_n
• | 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(I_n) = I_n
• 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.

☛Related Topics:

• Matrix Calculator
• Diagonal Matrix Calculator
• Transpose Matrix Calculator
• Matrix Addition Calculator

What is the Meaning of Matrix in Math?

A matrix in is an arrangement 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 matrix operations on them like addition, subtraction, multiplication, and so on. We have to take care of the orders while solving matrices.

• For the addition/subtraction of 2 matrices, their orders should be the same.
• For the multiplication of matrices, the number of columns of the left side matrix should be equal to the number of rows of the right side matrix.

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, and A^-1 = inverse matrix of A.

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?

The determinant formula for 2x2 matrix, \(A =\begin{pmatrix}a &b\\ \\c&d\end{pmatrix}\) is given by the formula |A| = 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 2 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
• (kA)^T= kA^T, k is any scalar
• (AB)^T= B^TA^T

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 by 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 a rectangular matrix
• Diagonal matrix
• Scalar matrix
• Identity matrix
• Null matrix
• Upper triangular matrix and lower triangular matrix
• Idempotent matrix
• Symmetric and skew-symmetric matrix

What are the Properties of Scalar Multiplication in Matrices?

Given the matrices A and B (both of the same order) and scalars K and l, the different properties associated with the multiplication of matrices can be given as,

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

What is a Matrix Polynomial?

Given a polynomial of the form, f(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-1+ . . . + a_n-1x + a_n, and A as a square matrix of order n. Then, f(A) = a₀Aⁿ + a₁A^n-1 + a₂A^n-2 + . . . + a_n-1A + a_nA + a_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 it is in either upper triangular or lower triangular form. To convert a matrix into the echelon form, we apply elementary row operations.

How to Express a Matrix as a Sum of Symmetric and Non-Symmetric Matrix?

Any square matrix A can be written as, A = P + Q, where P and Q are symmetric and skew-symmetric matrices respectively, such that, P = (A + A^T)/2 and Q = (A - A^T)/2.