Condition for Addition and Subtraction
If $A$ and $B$ are two matrices of the same order, then the addition operation is possible. The result of addition is a matrix of the same order and each of its elements is the sum of the two corresponding elements in $A$ and $B$.
If $A = (a_{ij})$ and $B = (b_{ij})$ are two matrices of order $m \times n$, then:
$A \pm B = (a_{ij} \pm b_{ij})$
The result is a matrix of the same order $m \times n$.
The Matrices:
If $A = \begin{pmatrix} 1 & -2 \\ 3 & 5 \\ 4 & 2 \end{pmatrix}$, $B = \begin{pmatrix} -1 & -2 \\ 5 & 1 \\ 3 & 2 \end{pmatrix}$ and $C = \begin{pmatrix} 2 & 0 & -1 \\ 3 & 4 & 6 \end{pmatrix}$
Find each of the following if it is possible:
Solution Steps:
(1) Finding $2A + C^t$ :
$2A + C^t = 2 \begin{pmatrix} 1 & -2 \\ 3 & 5 \\ 4 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 0 & -1 \\ 3 & 4 & 6 \end{pmatrix}^t$
$= \begin{pmatrix} 2 & -4 \\ 6 & 10 \\ 8 & 4 \end{pmatrix} + \begin{pmatrix} 2 & 3 \\ 0 & 4 \\ -1 & 6 \end{pmatrix} = \begin{pmatrix} 4 & -1 \\ 6 & 14 \\ 7 & 10 \end{pmatrix}$
(2) Finding $B + C$ :
It is impossible to add $B$ and $C$, because they don't have the same order. $B$ is $(3 \times 2)$ while $C$ is $(2 \times 3)$.
Properties of Matrix Addition
1. Closure property: The sum of any two matrices of the same order is a matrix of the same order.
2. Commutative property: $A + B = B + A$
3. Associative property: $(A + B) + C = A + (B + C)$
4. Additive Identity: The zero matrix $\mathcal{O}$ is the additive identity where $A + \mathcal{O} = \mathcal{O} + A = A$.
5. Additive Inverse: For any matrix $A$, there exists an additive inverse $-A$ such that $A + (-A) = \mathcal{O}$.
Remark:
Subtracting matrices operation is not commutative and not associative.
The Transpose Property
$(A \pm B)^T = A^T \pm B^T$
If $A = \begin{pmatrix} 2 & 3 \\ -1 & 5 \\ 6 & 7 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 4 \\ 2 & -3 \\ 4 & -8 \end{pmatrix}$, check that: $(A + B)^t = A^t + B^t$
Solution:
$\because A + B = \begin{pmatrix} 2 & 3 \\ -1 & 5 \\ 6 & 7 \end{pmatrix} + \begin{pmatrix} 0 & 4 \\ 2 & -3 \\ 4 & -8 \end{pmatrix} = \begin{pmatrix} 2 & 7 \\ 1 & 2 \\ 10 & -1 \end{pmatrix}$
$\therefore (A + B)^t = \begin{pmatrix} 2 & 1 & 10 \\ 7 & 2 & -1 \end{pmatrix} \quad \text{--- (1)}$
$\because A^t = \begin{pmatrix} 2 & -1 & 6 \\ 3 & 5 & 7 \end{pmatrix} \quad, \quad B^t = \begin{pmatrix} 0 & 2 & 4 \\ 4 & -3 & -8 \end{pmatrix}$
$\therefore A^t + B^t = \begin{pmatrix} 2 & -1 & 6 \\ 3 & 5 & 7 \end{pmatrix} + \begin{pmatrix} 0 & 2 & 4 \\ 4 & -3 & -8 \end{pmatrix} = \begin{pmatrix} 2 & 1 & 10 \\ 7 & 2 & -1 \end{pmatrix} \quad \text{--- (2)}$
Find the values of $a, b$ and $c$ that satisfy the equation:
$3 \begin{pmatrix} a & b \\ c & 3 \end{pmatrix} = 2 \begin{pmatrix} a & 6 \\ -1 & 3 \end{pmatrix} + \begin{pmatrix} 4 & b+4 \\ c+3 & 3 \end{pmatrix}$
Solution:
$\begin{pmatrix} 3a & 3b \\ 3c & 9 \end{pmatrix} = \begin{pmatrix} 2a & 12 \\ -2 & 6 \end{pmatrix} + \begin{pmatrix} 4 & b+4 \\ c+3 & 3 \end{pmatrix}$
$\therefore \begin{pmatrix} 3a & 3b \\ 3c & 9 \end{pmatrix} = \begin{pmatrix} 2a+4 & b+16 \\ c+1 & 9 \end{pmatrix}$
From equality of matrices:
If $A = \begin{pmatrix} 2 & 3 & -2 \\ -1 & 4 & 5 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & -1 & 3 \\ 5 & 2 & -4 \end{pmatrix}$, find the matrix $X$ that satisfies: $2[X^t - A] = 3B$
Solution:
$\because 2[X^t - A] = 3B \implies 2X^t - 2A = 3B$
$\therefore 2X^t = 3B + 2A$
$$ 2X^t = 3\begin{pmatrix} 0 & -1 & 3 \\ 5 & 2 & -4 \end{pmatrix} + 2\begin{pmatrix} 2 & 3 & -2 \\ -1 & 4 & 5 \end{pmatrix} $$
$$ 2X^t = \begin{pmatrix} 0 & -3 & 9 \\ 15 & 6 & -12 \end{pmatrix} + \begin{pmatrix} 4 & 6 & -4 \\ -2 & 8 & 10 \end{pmatrix} = \begin{pmatrix} 4 & 3 & 5 \\ 13 & 14 & -2 \end{pmatrix} $$
$\therefore X^t = \frac{1}{2}\begin{pmatrix} 4 & 3 & 5 \\ 13 & 14 & -2 \end{pmatrix} = \begin{pmatrix} 2 & \frac{3}{2} & \frac{5}{2} \\ \frac{13}{2} & 7 & -1 \end{pmatrix}$
Final Answer:
$$ X = (X^t)^t = \begin{pmatrix} 2 & \frac{13}{2} \\ \frac{3}{2} & 7 \\ \frac{5}{2} & -1 \end{pmatrix} $$
Frequently Asked Questions (FAQ)
Q1: Can we add two matrices with different orders?
No, the fundamental condition for matrix addition or subtraction is that both matrices must have the same order (same number of rows and columns).
Q2: Is matrix subtraction commutative?
No, matrix subtraction is NOT commutative ($A - B \neq B - A$). It is also NOT associative.
Q3: What happens when we add the zero matrix $\mathcal{O}$ to any matrix $A$?
The zero matrix $\mathcal{O}$ is the additive identity, so $A + \mathcal{O} = A$. The matrix remains unchanged.