Properties of Addition of Matrices will help for effective preparation of matrices. It is easy to perform addition operations using different properties. Different operations like addition, subtraction, multiplication, and inverse multiplication of matrices have different properties. Let us learn deeply about the Addition of Matrices Properties in this article.
Quickly know the output of the matrices after adding them by reading the entire article. The properties of matrix addition like Commutative, Associative, Additive Identity, and Additive Inverse of Matrix are explained in this article. Also, learn the Properties of multiplication of matrices on 10th Grade Math articles.
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What are the Properties of Matrix Addition? | Addition of Matrices Properties
There are four different Addition of Matrices Properties available. They are
- Commutative Law of Addition of Matrix
- Associative Law of Addition of Matrix
- Existence of Additive Identity of Matrix
- Existence of Additive Inverse of Matrix
Commutative Law of Addition of Matrix
The Addition of a Matrix is said to be commutative when the result of the matrix addition is equal to the addition of their interchanged position. If A and B are two matrices, then adding A and B and adding B and A must be equal.
A + B = B + A.
Proof: Let A = [aij]m × n and B = [bij]m × n where m is the number of rows and n is the number of columns of a matrix.
Now, add A and B. A + B = C which is also equal to [cij]m × n = aij + bij.
Then, add B and A. B + A = D = [dij]m × n which is also equal to [dij]m × n = bij + aij
Since C and D are of the same order and cij = dij then, C = D i.e., A + B = B + A.
From the definition of addition of matrices when they are commutative, cij = aij + bij = bij + aij = dij
Hence, the matrices A + B = B + A are commutative.
Associative Law of Addition of Matrix
The Associative Law of Addition of Matrix states that when any three matrices are added, then the grouping or association of the matrices does not affect the result. If A, B, and C are three matrices, then A + (B + C) = (A + B) + C.
Proof: Let A = [aij]m × n, B = [bij]m × n, C = [cij]m × n and where m is the number of rows and n is the number of columns of a matrix.
Let the addition of B and C is D.
Firstly, add B and C. D = B + C = [bij]m × n + [cij]m × n = [dij]m × n
Now, add A and D. Let the addition of A and D is P.
P = A + D = A + (B + C) = [aij]m × n + [dij]m × n = [pij]m × n
Let the addition of A and B is E.
Then, add A + B. E = A + B = [aij]m × n + [bij]m × n = [eij]m × n
Now, add E and C. Let the addition of E and C is Q.
Q + E + C = (A + B) + C = [aij]m × n + [bij]m × n = [qij]m × n
Since P and Q are of the same order and pij = qij then, P = Q.
Therefore, A + (B + C) = (A + B) + C.
Existence of Additive Identity of Matrix
The Additive Identity of Matrix is the matrix when added to a matrix with the same order results in the original matrix. When we add 0 matrix to any other matrix, then we get the same Matrix. If A is a matrix and O is a zero matrix, then adding A to O and O to A gives the same matrix A.
Proof: Let A = [aij]m × n and O = [oij]m × n where m is the number of rows and n is the number of columns of a matrix.
Now, add A and O. A + O = A which is also equal to [aij]m × n = aij + oij = aij = A.
Now, add O and A. O + A = A which is also equal to [aij]m × n = oij + aij = aij = A.
Here the null matrix is called the additive identity for the matrices.
Existence of Additive Inverse of Matrix
The additive inverse of a matrix is the matrix that when added to it, yields zero. The Additive Inverse of a matrix can be found by multiplying each element of the matrix by -1. If A is the matrix, then -A is the additive inverse of matrix A. Also, the additive inverse of matrix -A is -(-A) = A. A + (- A) = O = (- A) + A
Proof: Let A = [aij]m × n and the additive inverse of the matrix of A is -A which is equal to -[aij]m × n.
Now, add A and -A. A + (- A) = [aij] + [- aij] = [0] = O.
Now, add -A and A. -A + A = -[aij] + [ aij] = [0] = O.
Therefore, A + (- A) = O = (- A) + A. Here the matrix – A is called the additive inverse of the matrix A.
Transpose Property of Addition of Matrices
The addition property of transpose is defined as the sum of two transpose matrices that will be equal to the sum of the transpose of individual matrices.
The transpose of the sum of two matrices (A + B) is equal to the sum of the transposes of the respective matrices ATÂ + BT. (A + B)TÂ = ATÂ + BT
Determinant Property of Addition of Matrices
The determinant of the sum of two matrices |A + B| is equal to the sum of the determinants of the respective matrices |A| + |B|. |A + B| = |A| + |B|
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Properties of Matrix Addition with Examples
Matrix Addition Properties are easily understood by solving the below problems. Check out the below matrix problems which are solved those prove the addition properties of matrices.
Question 1.
If \( A =\left[
\begin{matrix}
3&5 \cr
7&9 \cr
\end{matrix}
\right]
\) and \( B =\left[
\begin{matrix}
4&6 \cr
2&1 \cr
\end{matrix}
\right]
\), then prove A and B matrices obey commutative property of addition.
Solution:
Given matrices are \( A =\left[
\begin{matrix}
3&5 \cr
7&9 \cr
\end{matrix}
\right]
\) and \( B =\left[
\begin{matrix}
4&6 \cr
2&1 \cr
\end{matrix}
\right]
\).
Both matrices are of the same order i.e, 2 × 2. Now, add the elements of matrix A with the respective elements of matrix B.
\( A + B =\left[
\begin{matrix}
3 + 4&5 + 6 \cr
7+ 2&9 + 1 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
7&11 \cr
9&10 \cr
\end{matrix}
\right]
\)
Now, add the elements of matrix B with the respective elements of matrix A.
\( B + A =\left[
\begin{matrix}
4 + 3&6 + 5 \cr
2 + 7&1 + 9 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
7&11 \cr
9&10 \cr
\end{matrix}
\right]
\)
Therefore, A + B = B + A. A and B matrices obey the commutative property of addition.
Example 2.
If \( A =\left[
\begin{matrix}
21&19&31 \cr
10&49&12 \cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
11&52&13 \cr
5&0&41 \cr
\end{matrix}
\right]
\), and \( C =\left[
\begin{matrix}
3&5&1 \cr
15&10&4 \cr
\end{matrix}
\right]
\) then prove A + (B + C) = (A + B) + C.
Solution:
Given that \( A =\left[
\begin{matrix}
21&19&31 \cr
10&49&12 \cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
11&52&13 \cr
5&0&41 \cr
\end{matrix}
\right]
\), and \( C =\left[
\begin{matrix}
3&5&1 \cr
15&10&4 \cr
\end{matrix}
\right]
\)
All the three matrices are of the same order i.e, 2 × 3.
Firstly, add B + C. Add the elements of matrix B with the respective elements of matrix C. B + C = \( \left[
\begin{matrix}
11&52&13 \cr
5&0&41 \cr
\end{matrix}
\right]
\) + \( \left[
\begin{matrix}
3&5&1 \cr
15&10&4 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
11 + 3&52 + 5&13 + 1 \cr
5 + 15&0 + 10&41 + 4 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
14&57&14 \cr
20&10&45 \cr
\end{matrix}
\right]
\)
Now, add A + (B + C). Add the elements of matrix A with the respective elements of matrix B + C.
A + (B + C) = \( \left[
\begin{matrix}
21&19&31 \cr
10&49&12 \cr
\end{matrix}
\right]
\) + \( \left[
\begin{matrix}
14&57&14 \cr
20&10&45 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
21 + 14&19 + 57&31 + 14 \cr
10 + 20&49 + 10&12 + 45 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
35&76&45 \cr
30&59&57 \cr
\end{matrix}
\right]
\)
Now, add A + B. Add the elements of matrix A with the respective elements of matrix B. A + B = \( \left[
\begin{matrix}
21&19&31 \cr
10&49&12 \cr
\end{matrix}
\right]
\) + \( \left[
\begin{matrix}
11&52&13 \cr
5&0&41 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
21 + 11&19 + 52&31 + 13 \cr
10 + 5&49 + 0&12 + 41 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
32&71&44 \cr
15&49&53 \cr
\end{matrix}
\right]
\)
Finally, add (A + B) + C. Add the elements of matrix A + B with the respective elements of matrix C. (A + B) + C = \( \left[
\begin{matrix}
32&71&44 \cr
15&49&53 \cr
\end{matrix}
\right]
\) + \( \left[
\begin{matrix}
3&5&1 \cr
15&10&4 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
32 + 3&71 + 5&44 + 1 \cr
15 + 15&49 + 10&53 + 4 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
35&76&45 \cr
30&59&57 \cr
\end{matrix}
\right]
\)
Therefore, A + (B + C) = (A + B) + C.
Example 3.
Find the Additive Identity of Matrix \( A =\left[
\begin{matrix}
91&72 \cr
16&24 \cr
\end{matrix}
\right]
\)
Solution:
Given matrix is \( A =\left[
\begin{matrix}
91&72 \cr
16&24 \cr
\end{matrix}
\right]
\)
Add the null matrix with the given matrix A.
\( \left[
\begin{matrix}
91&72 \cr
16&24 \cr
\end{matrix}
\right]
\) + \( \left[
\begin{matrix}
0&0 \cr
0&0 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
91 + 0&72 + 0 \cr
16 + 0&24 + 0 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
91&72 \cr
16&24 \cr
\end{matrix}
\right]
\) = A
Therefore, A + O = A.
Example 4.
If the given matrix \( A =\left[
\begin{matrix}
91&72 \cr
16&24 \cr
\end{matrix}
\right]
\) obeys an Additive Inverse of Matrix, find the inverse of the matrix A.
Solution:
Given matrix is \( A =\left[
\begin{matrix}
91&72 \cr
16&24 \cr
\end{matrix}
\right]
\) obeys an Additive Inverse of Matrix.
Multiply the given matrix A with -1 to get the -A.
(-1) \( \left[
\begin{matrix}
91&72 \cr
16&24 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
-91&-72 \cr
-16&-24 \cr
\end{matrix}
\right]
\) = -A.
\( \left[
\begin{matrix}
91&72 \cr
16&24 \cr
\end{matrix}
\right]
\) + \( \left[
\begin{matrix}
-91&-72 \cr
-16&-24 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
0&0 \cr
0&0 \cr
\end{matrix}
\right]
\)
A + (-A) = 0.
Therefore, \(-A \left[
\begin{matrix}
-91&-72 \cr
-16&-24 \cr
\end{matrix}
\right]
\)
FAQs on Matrix Addition Properties
1. What is the Commutative Law of Addition of Matrices?
If A and B are two matrices, then A + B = B + A is the Commutative Law of Addition of Matrices.
2. What are the Properties of Matrix Addition?
The properties of Matrix Addition are Commutative Law, Associative Law, Additive Identity of Matrix, and Additive Inverse of Matrix.
3. What is the Addition of Matrices?
The addition of matrices is performing an addition operation on two matrices.
4. What is the additive identity of a?
Additive identity of a is 0, since a + 0 = a.
Summary
Properties of Matrix Addition are given in this article. Students can easily solve the matrix addition problems using the given properties and find the output. Learn the properties of the addition of matrices and also their proofs to know the addition process of matrices. Get connected with our website to know updates according to the new syllabus.