Properties of Matrices
Properties of matrices are helpful in performing numerous operations involving two or more matrices. The algebraic operations of addition, subtraction, multiplication, inverse multiplication of matrices, and involving different types of matrices can be easily performed by the use of properties of matrices. The additive, multiplicative identity, and inverse of matrices are also included in this study of properties of matrices.
Let us learn more about the properties of matrix addition, properties of scalar multiplication of matrices, properties of matrix multiplication, properties of transpose matrix, properties of an inverse matrix with examples and frequently asked questions.
What Are the Properties of Matrices?
The properties of matrices help in performing numerous operations on matrices. The properties of matrices can be broadly classified into the following five properties.
 Properties of Matrix Addition
 Properties of Scalar Multiplication of Matrix
 Properties of Matrix Multiplication
 Properties of Transpose Matrix
 Properties of Inverse Matrix and other properties.
Let us check more about each of the properties of matrices.
Properties of Matrix Addition
The addition of matrices satisfies the following properties of matrices.
 Commutative Law. For the given two matrixes, matrix A and matrix B of the same order, say m x n, then A + B = B + A.
 Associative law: For any three matrices, A , B, C of the same order m x n, we have (A + B) + C = A + (B + C)
 Existence of additive identity Let A be a matrix of order m × n, and O be a zero matrix or a null matrix of the same order m × n , then A + O = O + A = A. In other words, O is the additive identity for matrix addition.
 Existence of additive inverse Let A be a matrix of order m × n. and let A be another matrix of order m × n such that A + (– A) = (– A) + A= O. So the matrix – A is the additive inverse of A or the negative of matrix A.
Properties of Scalar Multiplication of Matrix
The properties of scalar multiplication of matrix involve a scalar constant and a matrix. For matrixes A and B of order m x n, and k and l as scalars values, the property of scalar multiplication of matrices is as follows.
 The product of a constant with the sum of matrices is equal to the sum of the individual product of the constant and the matrix. k(A + B) = kA + kB
 The product of the sum of the constants with a matrix is equal to the sum of the product of each of the constants with the matrix. (k + l)A = kA + lA
Here both the matrix A and B are of the same order and the constants K and l are any real number values.
Properties of Multiplication of Matrices
The following properties of matrix multiplication help in performing numerous operations involving matrix multiplication. The condition for matrix multiplication is the number of columns in the first matrix should be equal to the number of rows in the second matrix. Let us check the three important properties of matrices.
 Associative Property: For any three matrices A, B, C following the matrix multiplication conditions, we have (AB)C = A(BC). Here both sides of the matrix multiplication are defined.
 Distributive Property: For any three matrices A, B, C following the matrix multiplication conditions, we have A(B + C) = AB + AC.
 The existence of multiplicative identity. For a square matrix A, having the order m × n, and an identity matrix I of the same order we have AI = IA = A. Here the product of the identity matrix with the given matrix results in the same matrix.
Properties of Transpose of a Matrix
The properties of matrices for matrices A and B of the same order m × n, and a constant k is defined. The following are some of the important properties of the transpose of a matrix.
 The transpose of a matrix on further taking a transpose for the second time results in the original matrix. (A')' = A
 The transpose of the product of a constant and a matrix is equal to the product of the constant and the transpose of the matrix. (kA)' = kA'
 The transpose of the sum of two matrices is equal to the sum of the transpose of the individual matrices. (A + B)' = A' + B'
 The transpose of the product of two matrices is equal to the product of the transpose of the second matrix and the transpose of the first matrix. (AB)' = B'A'
Other Important Properties of Matrices
In addition to the above set of properties of matrices, some of the other important properties have been grouped and presented across the below points.
 For a square matrix with real number entries, A + A' is a symmetric matrix, and A  A' is a skewsymmetric matrix.
 A square matrix can be expressed as a sum of a symmetric and skewsymmetric matrix. A = 1/2(A + A') + 1/2(A  A').
 The inverse of a matrix if it exists is unique. AB = BA = I.
 If matrix A is the inverse of matrix B, then matrix B is the inverse of matrix A.
 If A and B are invertible matrices of the same order m × n, then (AB)^{1} = B^{1}A^{1}.
Related Topics
The following topics would help in a better understanding of the properties of matrices.
Examples on Properties of Matrices

Example 1: For the matrix \(\begin{pmatrix}4&3\\2&1\end{pmatrix}\) prove the transpose property of (A')' = A.
Solution:
Let the given matrix be A = \(\begin{pmatrix}4&3\\2&1\end{pmatrix}\)
The transpose of the matrix is A' = \(\begin{pmatrix}4&3\\2&1\end{pmatrix}\)
The transpose of the transpose matrix is (A')' = \(\begin{pmatrix}4&3\\2&1\end{pmatrix}\)
This on observation is equal to the origin matrix A, and hence it satisfies the matrix transpose property of (A')' = A.
Therefore, the given matrix satisfies the matrix transpose property of (A')' = A.
Example 2: Prove that the matrixes A =\(\begin{pmatrix}6&1\\0&2\end{pmatrix}\), B = \(\begin{pmatrix}4&5\\3&2\end{pmatrix}\), C = \(\begin{pmatrix}3&4\\4&2\end{pmatrix}\), follow the distributive property of matrix multiplication.
Solution:
The given matrices are A =\(\begin{pmatrix}6&1\\0&2\end{pmatrix}\), B = \(\begin{pmatrix}4&5\\3&2\end{pmatrix}\), C = \(\begin{pmatrix}3&4\\4&2\end{pmatrix}\).
And the distributive property of matrix multiplication is A(B + C) = AB + AC. Let us first find the product AB and AC.
AB = \(\begin{pmatrix}6&1\\0&2\end{pmatrix}\) ×\(\begin{pmatrix}4&5\\3&2\end{pmatrix}\)
= \(\begin{pmatrix}6×4+1×3&6×5+1×2\\0×4+2×3&0×5+2×2\end{pmatrix}\) = \(\begin{pmatrix}27&32\\6&4\end{pmatrix}\)
AC = \(\begin{pmatrix}6&1\\0&2\end{pmatrix}\) × \(\begin{pmatrix}3&4\\4&2\end{pmatrix}\)
= \(\begin{pmatrix}6×(3) + 1 × 4&6×4 + 1 × 2\\0×(3) + 2×4&0×4+2×2\end{pmatrix}\) = \(\begin{pmatrix}14&26\\8&4\end{pmatrix}\)
B + C = \(\begin{pmatrix}4&5\\3&2\end{pmatrix}\) + \(\begin{pmatrix}3&4\\4&2\end{pmatrix}\)
= \(\begin{pmatrix}4+(3)&5+4\\3+4&2+2\end{pmatrix}\) = \(\begin{pmatrix}1&9\\7&4\end{pmatrix}\)
A(B + C) = \(\begin{pmatrix}6&1\\0&2\end{pmatrix}\) × \(\begin{pmatrix}1&9\\7&4\end{pmatrix}\)
= \(\begin{pmatrix}6×1+1×7&6×9+1×4\\0×1+2×7&0×9+2×4\end{pmatrix}\) = \(\begin{pmatrix}13&58\\14&8\end{pmatrix}\)
AB + AC = \(\begin{pmatrix}27&32\\6&4\end{pmatrix}\) + \(\begin{pmatrix}14&26\\8&4\end{pmatrix}\)
= \(\begin{pmatrix}27 + (14)&32+26\\6+8&4+4\end{pmatrix}\) = \(\begin{pmatrix}13&58\\14&8\end{pmatrix}\)
From the above two expressions we can observe that A(B + C) = AB + AC.
Therefore, the given matrices follow the distributive property of matrix multiplication.
FAQs on Properties of Matrices
What Are the Addition Properties of Matrices?
The addition of matrices satisfies the following properties of matrices.
 Commutative Law. For the given two matrixes, A + B = B + A.
 Associative law: For any three matrices, A , B, C, we have (A + B) + C = A + (B + C)
 Existence of additive identity Let A be a matrix of order m × n, and O be a zero matrix or a null matrix of the same order m × n , then A + O = O + A = A.
 Existence of additive inverse Let A be a matrix of order m × n. and let A be another matrix of order m × n such that A + (– A) = (– A) + A= O.
What Are Properties of Transpose of Matrices?
For given two matrices, A and B, the properties of the transpose 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 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?
For 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 is as follows.
 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
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.
What Are the Multiplication Properties of Matrices
The following properties of matrix multiplication help in performing numerous operations involving matrix multiplication.
 Associative Property: For any three matrices A, B, C we have (AB)C = A(BC).
 Distributive Property: For any three matrices A, B, C we have A(B + C) = AB + AC.
 The existence of multiplicative identity. For a square matrix A, having the order m × n, and an identity matrix I of the same order we have AI = IA = A.
What Are the Properties of Matrices for Inverse of a Matrix?
The following are the important properties of the inverse of a matrix.
 The inverse of a matrix if it exists is unique. AB = BA = I.
 If matrix A is the inverse of matrix B, then matrix B is the inverse of matrix A.
 If A and B are invertible matrices of the same order m × n, then (AB)^{1} = B^{1}A^{1}.
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