Addition of Matrices

A matrix is a rectangular arrangement/array of numbers, symbols, or expressions, structured in rows and columns. It is a powerful mathematical tool used in various fields such as algebra, statistics, physics, and engineering to represent and manipulate data.

Matrix addition involves combining two or more matrices by adding their corresponding elements. Unlike the arithmetic addition of numbers, matrix addition requires that the matrices have the same dimensions.

1.0Introduction to Matrices

A matrix is a rectangular array of numbers, expressions, or symbols organized in rows and columns. Horizontal rows are denoted by “m” and vertical columns by “n,” so a matrix of size m × n has m rows and n columns.

2.0Addition of Matrices

Matrix addition is a fundamental operation where corresponding elements of two matrices are added together to form a new matrix of the same dimensions. 

Matrix Addition and Subtraction

To add or subtract matrices, ensure they have the same dimensions. For example, given matrices A and B:

$A=\left[\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right],B=\left[\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]$

Their sum is:

$A+B=\left[\begin{array}{ll}{a}_{11}+b_{11}& a_{12}+b_{12}\\ a_{21}+b_{21}& a_{22}+b_{22}\end{array}\right]$

Similarly, their difference is:

$A-B=\left[\begin{array}{ll}{a}_{11}-b_{11}& a_{12}-b_{12}\\ a_{21}-b_{21}& a_{22}-b_{22}\end{array}\right]$

Matrix addition and subtraction are foundational operations that obey specific rules and properties, facilitating more complex matrix manipulations.

Addition of 3 × 3 Matrices

To add two 3 × 3 matrices, add each element from one matrix to the corresponding element in the other matrix:

$\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]+\left[\begin{array}{lll}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right]=\left[\begin{array}{lll}{a}_{11}+b_{11}& a_{12}+b_{12}& a_{13}+b_{13}\\ a_{21}+b_{21}& a_{22}+b_{22}& a_{23}+b_{23}\\ a_{31}+b_{31}& a_{32}+b_{32}& a_{33}+b_{33}\end{array}\right]$

3.0Properties of Matrix Addition with Examples

Matrix addition has several important properties that make it a fundamental operation in linear algebra. Here are the key properties:

1. Commutative Property

Matrix addition is commutative, which means that the order of addition does not affect the result. If A and B are matrices of the same dimension, then:

A + B = B + A 

2. Associative Property

Matrix addition is associative, which means that when adding three or more matrices, the way in which the matrices are grouped does not affect the result. If A, B, and C are matrices of the same dimension, then:

(A + B) + C = A + (B + C) 

3. Additive Identity

There exists an additive identity matrix (also called the zero matrix) that, when added to any matrix A, results in the matrix A itself. The zero matrix 0 has all its elements equal to zero. For any matrix A:

A + 0 = A

4. Additive Inverse

For every matrix A, there exists an additive inverse matrix –A, such that:

A + (–A) = 0 

Here, –A is the matrix where each element is the negation(negative) of the corresponding element in A.

5. Compatibility with Scalar Multiplication

When a matrix is multiplied by a scalar, the resulting matrix can still be added to another matrix of the same dimension. If A and B are matrices and k represent a scalar, then:

$k\cdot (A+B)=k\cdot A+k\cdot B$

Commutative Property Example

Let $A=\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{ll}5& 6\\ 7& 8\end{array}\right]$

then, 

A+B= $\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{ll}5& 6\\ 7& 8\end{array}\right]=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]$

$B+A=\left[\begin{array}{ll}5& 6\\ 7& 8\end{array}\right]+\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]$

This demonstrates that A + B = B + A.

Associative Property Example

Let $A=\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{ll}5& 6\\ 7& 8\end{array}\right],andC=\left[\begin{array}{cc}9& 10\\ 11& 12\end{array}\right].$

$(A+B)+C=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]+\left[\begin{array}{cc}9& 10\\ 11& 12\end{array}\right]=\left[\begin{array}{ll}15& 18\\ 21& 24\end{array}\right]$

$A+(B+C)=\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{ll}14& 16\\ 18& 20\end{array}\right]=\left[\begin{array}{ll}15& 18\\ 21& 24\end{array}\right]$

This demonstrates that (A + B) + C = A + (B + C).

These properties ensure that matrix addition behaves in a predictable and consistent manner, allowing for various applications in mathematical operations and transformations.

4.0Rule for Matrix Addition

Matrix addition is possible only when the matrices have the same dimensions. The rule is to add corresponding elements from each matrix.

Can You Add a 3 × 2 and a 2 × 3 Matrix?

No, you cannot add a 3 × 2 matrix and a 2 × 3 matrix because they do not have the same dimensions.

Addition of Matrices with Variables

When matrices contain variables, you still add corresponding elements, treating the variables as algebraic terms. For example:

$\left[\begin{array}{ll}x& y\\ z& w\end{array}\right]+\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]=\left[\begin{array}{ll}x+a& y+b\\ z+c& w+d\end{array}\right]$

Matrix addition involves combining two or more matrices by adding their corresponding elements. Unlike arithmetic addition of numbers, matrix addition follows specific rules. The matrices must have the same dimensions to be added.

5.0Addition of Matrix Solved Example

Example 1: Let $A(x)=\left[\begin{array}{cc}x& 2x\\ 3x& 4x\end{array}\right]$ and $B(x)=\left[\begin{array}{ll}5x& 6x\\ 7x& 8x\end{array}\right]$. Find A(x) + B(x) and evaluate at x =2.

Solution:

$A(x)=\left[\begin{array}{cc}x& 2x\\ 3x& 4x\end{array}\right],B(x)=\left[\begin{array}{cc}5x& 6x\\ 7x& 8x\end{array}\right]$

$A(x)=\left[\begin{array}{cc}x& 2x\\ 3x& 4x\end{array}\right],B(x)=\left[\begin{array}{cc}5x& 6x\\ 7x& 8x\end{array}\right]$

$A(x)+B(x)=\left[\begin{array}{cc}x+5x& 2x+6x\\ 3x+7x& 4x+8x\end{array}\right]=\left[\begin{array}{cc}6x& 8x\\ 10x& 12x\end{array}\right]$

At x = 2, $A(x)+B(x)=\left[\begin{array}{cc}6(2)& 8(2)\\ 10(2)& 12(2)\end{array}\right]=\left[\begin{array}{ll}12& 16\\ 20& 24\end{array}\right]$

Example 2: Let $A=\left[\begin{array}{lll}5& 0& -2\\ 3& 2& -7\end{array}\right]$and $B=\left[\begin{array}{ccc}4& -3& -6\\ -1& 0& 4\end{array}\right]$. Then find A +B.

Solution:

$A=\left[\begin{array}{lll}5& 0& -2\\ 3& 2& -7\end{array}\right]$ and $B=\left[\begin{array}{ccc}4& -3& -6\\ -1& 0& 4\end{array}\right].$

Clearly, each one of A and B is a 2 × 3 matrix.

So, A and B are comparable matrices.

∴ A + B is defined.

Now, $A+B=\left[\begin{array}{ccc}5+4& 0+(-3)& -2+(-6)\\ 3+(-1)& 2+0& -7+4\end{array}\right]=\left[\begin{array}{ccc}9& -3& -8)\\ 2& 2& -3\end{array}\right]$

Example 3: If $A=\left[\begin{array}{ccc}3& -2& 0\\ -5& 7& \sqrt{2}\end{array}\right]$, find (–A) and verify that A + (–A) = (–A) + A = 0

Solution:

Clearly, we have :

$(-A)=\left[\begin{array}{ccc}-3& 2& 0\\ 5& -7& -\sqrt{2}\end{array}\right]$

$A+(-A)=\left[\begin{array}{ccc}3& -2& 0\\ -5& 7& \sqrt{2}\end{array}\right]+\left[\begin{array}{ccc}-3& 2& 0\\ 5& -7& -\sqrt{2}\end{array}\right]$

$=\left[\begin{array}{ccc}3+(-3)& -2+2& 0+0\\ -5+5& 7+(-7)& \sqrt{2}+(-\sqrt{2})\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]=0$

And, $(-A)+A=\left[\begin{array}{ccc}-3& 2& 0\\ 5& -7& -\sqrt{2}\end{array}\right]+\left[\begin{array}{ccc}3& -2& 0\\ -5& 7& \sqrt{2}\end{array}\right]$

$=\left[\begin{array}{ccc}-3+3& 2+(-2)& 0+0\\ 5+(-5)& -7+7& -\sqrt{2}+\sqrt{2}\end{array}\right]=\left[\begin{array}{lll}0& 0& 0\\ 0& 0& 0\end{array}\right]$

Hence, A + (–A) = (–A) + A = O.

Example 4: Let $A=\left[\begin{array}{ccc}2& 3& 5\\ -1& 0& 4\end{array}\right]$ and $B=\left[\begin{array}{ccc}4& -2& 3\\ 2& 6& -1\end{array}\right]$. Verify that A + B = B + A.

Solution:

Here, A is a 2 × 3 matrix and B is a 2 × 3 matrix. So, A and B are comparable.

Therefore, (A + B) and (B + A) both exist and each is a 2 × 3 matrix.

Now, $A+B=\left[\begin{array}{ccc}2& 3& 5\\ -1& 0& 4\end{array}\right]+\left[\begin{array}{ccc}4& -2& 3\\ 2& 6& -1\end{array}\right]$

$=\left[\begin{array}{ccc}2+4& 3+(-2)& 5+3\\ -1+2& 0+6& 4+(-1)\end{array}\right]=\left[\begin{array}{lll}6& 1& 8\\ 1& 6& 3\end{array}\right]$ .

And, $B+A=\left[\begin{array}{ccc}4& -2& 3\\ 2& 6& -1\end{array}\right]+\left[\begin{array}{ccc}2& 3& 5\\ -1& 0& 4\end{array}\right]$

$=\left[\begin{array}{ccc}4+2& -2+3& 3+5\\ 2+(-1)& 6+0& (-1)+4\end{array}\right]=\left[\begin{array}{lll}6& 1& 8\\ 1& 6& 3\end{array}\right]$

Hence, A + B = B + A.

Example 5: If $\left[\begin{array}{cc}a+4& 3b\\ 8& -6\end{array}\right]=\left[\begin{array}{cc}2a+2& b+2\\ 8& a-8b\end{array}\right]$, then find the value of (a – 2b).

Solution:

Comparing the corresponding elements the given equal matrices, we have a + 4 = 2a + 2, 3b = b + 2 and a – 8b = –6.

From these equations, we get a = 2 and b = 1.

∴ (a – 2b) = (2 – 2 × 1) = (2 – 2) = 0.

Example 6: If $\left[\begin{array}{ccc}9& -1& 4\\ -2& 1& 3\end{array}\right]=A+\left[\begin{array}{ccc}1& 2& -1\\ 0& 4& 9\end{array}\right]$, then find the matrix A.

Solution:

Clearly, we have 

$A=\left[\begin{array}{ccc}9& -1& 4\\ -2& 1& 3\end{array}\right]-\left[\begin{array}{ccc}1& 2& -1\\ 0& 4& 9\end{array}\right]$

$=\left[\begin{array}{ccc}9-1& -1-2& 4+1\\ -2-0& 1-4& 3-9\end{array}\right]=\left[\begin{array}{ccc}8& -3& 5\\ -2& -3& -6\end{array}\right]$

6.0Addition of Matrix Practice Questions

1. Given matrices $A=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$and $B=\left[\begin{array}{ll}e& f\\ g& h\end{array}\right]$, find A + B and 2A + 3B.
2. For matrices A and B of dimensions 3 × 3, where $A=\left[\begin{array}{lll}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{lll}9& 8& 7\\ 6& 5& 4\\ 3& 2& 1\end{array}\right]$, compute A + B.
3. Given matrices A and B and a polynomial P(x) = A + xB, find P(x) and evaluate P(x) at x = 1 and x = –1.
4. If $A=\left[\begin{array}{cc}3& 5\\ -2& 0\\ 6& -1\end{array}\right],B=\left[\begin{array}{cc}-1& -3\\ 4& 2\\ -2& 3\end{array}\right]$ and $C=\left[\begin{array}{cc}0& 2\\ 3& -4\\ 1& 6\end{array}\right]$. Then verify that (A + B) + C = A + (B + C).
5. Let $A=\left[\begin{array}{ll}2& 4\\ 3& 2\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]$ and $C=\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$. Find: 

• A + 2B
• 2C + A