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Matrices also follow the certain properties such as whole numbers and integers. Generally, there are four properties of a matrix in terms of addition and scalar multiplication as follows:
1. Commutative property
2. Associative property
3. Scalar identity for unit matrix
4. Distributive property
Let's dive into each property individually with an example.
Let A, B, C be m×n matrices and p and q be two non-zero scalars (numbers). Then we have the following properties.
Commutative property of matrix addition:
Changing the order of the matrices does not change the result of the matrices.
A + B = B + A
Example:
Consider the matrices A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix} then verify that A + B = B + A
Solution:
First, we find the sum of A + B matrices.
A + B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}+ \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1+5 & 2+6\\ 3+7 & 4+8 \end{bmatrix}=\begin{bmatrix} 6& 8\\ 10& 12 \end{bmatrix}………..(1)
Similarly, let's find B + A.
B + A = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}+ \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 5+1 & 6+2\\ 7+3 & 8+4 \end{bmatrix}=\begin{bmatrix} 6& 8\\ 10& 12 \end{bmatrix}………..(2)
From (1), (2) we can see that A + B = B + A. Thus, the given matrices are satisfy the Commutative property of matrices.
Associative property:
1. Associative property of matrix addition:
It says that rearranging parenthesis in the matrix expression will not change the result of the matrices.
(A + B)+ C = A + (B+C)
Example:
If A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix}, C = \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix} then verify that (A + B)+ C = A + (B+C).
Solution:
First we find the sum of A + B matrices then add it's result with the \(C) matrix.
(A + B) + C= \left (\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}+ \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix}\right ) + \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}
=\begin{bmatrix} 1+5 & 2+6\\ 3+7 & 4+8 \end{bmatrix}+ \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}
=\begin{bmatrix} 6 & 8\\ 10 & 12 \end{bmatrix}+ \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}
=\begin{bmatrix} 6+9 & 8+10\\ 10+11 & 12+12 \end{bmatrix}= \begin{bmatrix} 15 & 18\\ 21 & 24 \end{bmatrix}……….(1)
Similarly, let's find A + (B + C).
A + ( B + C) =\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}+ \begin{bmatrix} 5+9 & 6+10\\ 7+11 & 8+12 \end{bmatrix}
=\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}+ \begin{bmatrix} 14 & 16\\ 18 & 20 \end{bmatrix}
= \begin{bmatrix} 1+14 & 2+16\\ 18+3 & 4+20 \end{bmatrix}=\begin{bmatrix} 15 & 18\\ 21 & 24 \end{bmatrix}……….(2)
2. Associative property of scalar multiplication - (pq)A = p(Aq)
Example:
Verify the associative property if A = \begin{bmatrix} 2 & 4\\ 6 & 8 \end{bmatrix} and p = 4 and q = 6.
Solution:
We know the associative property of scalar multiplication is (pq)A = p(Aq). Here, p = 4 and q = 6.
So, (4 × 6) A = (4 × 6) \begin{bmatrix} 2 & 4\\ 6 & 8 \end{bmatrix}
= 24\begin{bmatrix} 2 & 4\\ 6 & 8 \end{bmatrix} = \begin{bmatrix} 48 & 96\\ 144 & 192 \end{bmatrix}……….(1)
Similarly, let's find 4( A × 6) = 4 (6 ×\begin{bmatrix} 2 & 4\\ 6 & 8 \end{bmatrix})
= 4 × \begin{bmatrix} 12 & 24\\ 36 & 48 \end{bmatrix} = \begin{bmatrix} 48 & 96\\ 144 & 192 \end{bmatrix}……..(2)
From the equations (1) and (2), we can see that (pq)A = p(Aq).
Hence, the given matrix follows the associative property of scalar multiplication.
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