# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/matrix/matrix-addition Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/matrix/matrix-addition/en.mdx Learn matrix addition with same-order matrices. Understand properties like commutativity and associativity, then solve practical problems with worked examples. --- ## What Is Matrix Addition? Matrix addition is a fundamental operation in matrix algebra where two or more matrices are combined to produce a new matrix. This operation can only be performed if the matrices being added have the same size or order. The result of the addition is a new matrix of the same order, where each element is the sum of the corresponding elements (elements in the same position) from the original matrices. ## Formal Definition of Matrix Addition Two matrices, let's say matrix $$A$$ and matrix $$B$$, can be added if and only if both matrices have the same order. Visible text: Two matrices, let's say matrix and matrix , can be added if and only if both matrices have the same order. Suppose matrix $$A$$ is of order $$m \times n$$ with elements $$a_{ij}$$ (element in the $$i$$-th row and $$j$$-th column), and matrix $$B$$ is also of order $$m \times n$$ with elements $$b_{ij}$$. Visible text: Suppose matrix is of order with elements (element in the -th row and -th column), and matrix is also of order with elements . Then, the sum of matrix $$A$$ and matrix $$B$$, which we call matrix $$C$$, is written as $$C = A + B$$. Matrix $$C$$ will also be of order $$m \times n$$, with elements $$c_{ij}$$ defined as: Visible text: Then, the sum of matrix and matrix , which we call matrix , is written as . Matrix will also be of order , with elements defined as: ```math c_{ij} = a_{ij} + b_{ij} ``` This means that each element in the resulting matrix is obtained by adding the elements that are in the same position from the two matrices being added. ## How to Perform Matrix Addition To add two matrices, follow these steps: 1. **Ensure Same Order**: Check if both matrices have the same number of rows and columns. If not, addition cannot be performed. 2. **Add Corresponding Elements**: Add the elements that are in the same row and column position from both matrices. 3. **Form the Resultant Matrix**: Arrange the sums of these elements into a new matrix of the same order. Visible text: 1. **Ensure Same Order**: Check if both matrices have the same number of rows and columns. If not, addition cannot be performed. 2. **Add Corresponding Elements**: Add the elements that are in the same row and column position from both matrices. 3. **Form the Resultant Matrix**: Arrange the sums of these elements into a new matrix of the same order. ### Example of Matrix Addition Suppose we have two matrices, $$P$$ and $$Q$$, as follows: Visible text: Suppose we have two matrices, and , as follows: Component: MathContainer Children: ```math P = \begin{bmatrix} 1 & 5 \\ -2 & 0 \\ 4 & 7 \end{bmatrix} ``` ```math Q = \begin{bmatrix} 6 & -3 \\ 8 & 1 \\ -2 & 9 \end{bmatrix} ``` Both matrices are of order $$3 \times 2$$ ($$3$$ rows and $$2$$ columns), so they can be added. Visible text: Both matrices are of order ( rows and columns), so they can be added. Then, $$P+Q$$ is: Visible text: Then, is: Component: MathContainer Children: ```math P+Q = \begin{bmatrix} 1 & 5 \\ -2 & 0 \\ 4 & 7 \end{bmatrix} + \begin{bmatrix} 6 & -3 \\ 8 & 1 \\ -2 & 9 \end{bmatrix} ``` ```math = \begin{bmatrix} 1+6 & 5+(-3) \\ -2+8 & 0+1 \\ 4+(-2) & 7+9 \end{bmatrix} ``` ```math = \begin{bmatrix} 7 & 2 \\ 6 & 1 \\ 2 & 16 \end{bmatrix} ``` Thus, the sum of matrix $$P$$ and matrix $$Q$$ is the matrix $$\begin{bmatrix} 7 & 2 \\ 6 & 1 \\ 2 & 16 \end{bmatrix}$$. Visible text: Thus, the sum of matrix and matrix is the matrix . ### Matrices That Cannot Be Added Suppose matrix $$X = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ and matrix $$Y = \begin{bmatrix} 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix}$$. Visible text: Suppose matrix and matrix . Matrix $$X$$ is of order $$2 \times 2$$, while matrix $$Y$$ is of order $$2 \times 3$$. Since their orders are different, matrix $$X$$ and matrix $$Y$$ cannot be added. Visible text: Matrix is of order , while matrix is of order . Since their orders are different, matrix and matrix cannot be added. ## Properties of Matrix Addition Matrix addition has several important properties, similar to the properties of addition for real numbers. Let $$A$$, $$B$$, and $$C$$ be matrices of the same order, and $$O$$ be the zero matrix (a matrix where all elements are zero) of the same order as $$A$$, $$B$$, and $$C$$. Visible text: Matrix addition has several important properties, similar to the properties of addition for real numbers. Let , , and be matrices of the same order, and be the zero matrix (a matrix where all elements are zero) of the same order as , , and . 1. **Commutative Property**: The order of matrix addition does not affect the result. ```math A + B = B + A ``` This means that adding matrix $$A$$ to $$B$$ will produce the same matrix as adding matrix $$B$$ to $$A$$. 2. **Associative Property**: The grouping in the addition of three or more matrices does not affect the result. ```math (A + B) + C = A + (B + C) ``` This means you can add $$A$$ and $$B$$ first, then add the result to $$C$$, or add $$B$$ and $$C$$ first, then add $$A$$ to the result. The final outcome will be the same. 3. **Existence of an Identity Element (Zero Matrix)**: There exists a zero matrix $$O$$ that acts as the identity element in addition. ```math A + O = O + A = A ``` This means that if a matrix is added to a zero matrix (of the same order), the result is the matrix itself. This zero matrix plays a role similar to the number $$0$$ in the addition of numbers. 4. **Existence of an Additive Inverse (Opposite of a Matrix)**: Every matrix $$A$$ has an additive inverse, denoted as $$-A$$, which when added to $$A$$ results in the zero matrix $$O$$. ```math A + (-A) = O ``` The matrix $$-A$$ is a matrix where each element is the opposite (negative) of the corresponding elements of matrix $$A$$. For example, if $$a_{ij}$$ is an element of $$A$$, then $$-a_{ij}$$ is an element of $$-A$$. Visible text: 1. **Commutative Property**: The order of matrix addition does not affect the result. This means that adding matrix to will produce the same matrix as adding matrix to . 2. **Associative Property**: The grouping in the addition of three or more matrices does not affect the result. This means you can add and first, then add the result to , or add and first, then add to the result. The final outcome will be the same. 3. **Existence of an Identity Element (Zero Matrix)**: There exists a zero matrix that acts as the identity element in addition. This means that if a matrix is added to a zero matrix (of the same order), the result is the matrix itself. This zero matrix plays a role similar to the number in the addition of numbers. 4. **Existence of an Additive Inverse (Opposite of a Matrix)**: Every matrix has an additive inverse, denoted as , which when added to results in the zero matrix . The matrix is a matrix where each element is the opposite (negative) of the corresponding elements of matrix . For example, if is an element of , then is an element of . ## Exercises **Problem** $$1$$ Visible text: **Problem** Given the following matrices: Component: MathContainer Children: ```math A = \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix} ``` ```math B = \begin{bmatrix} -2 & 9 & -4 \\ 6 & -3 & 1 \end{bmatrix} ``` ```math C = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} ``` Calculate $$A+B$$ and $$B+A$$. Then, determine if $$A+C$$ can be calculated and provide your explanation. Visible text: Calculate and . Then, determine if can be calculated and provide your explanation. **Problem** $$2$$ Visible text: **Problem** Determine the values of $$x, y,$$ and $$z$$ from the following matrix addition: Visible text: Determine the values of and from the following matrix addition: ```math \begin{bmatrix} 2x & 5 \\ -1 & 3y \end{bmatrix} + \begin{bmatrix} 4 & -2 \\ z & 7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix} ``` **Problem** $$3$$ Visible text: **Problem** If $$P = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix}$$, determine the matrix $$-P$$ (the additive inverse of $$P$$) and prove that $$P + (-P) = O$$, where $$O$$ is the zero matrix of the same order. Visible text: If , determine the matrix (the additive inverse of ) and prove that , where is the zero matrix of the same order. ### Answer Key **Problem** $$1$$ Visible text: **Problem** Given matrices: Component: MathContainer Children: ```math A = \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix} ``` ```math B = \begin{bmatrix} -2 & 9 & -4 \\ 6 & -3 & 1 \end{bmatrix} ``` ```math C = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} ``` Addition of matrix $$A$$ and $$B$$ ($$A+B$$): Visible text: Addition of matrix and (): Component: MathContainer Children: ```math A+B = \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix} + \begin{bmatrix} -2 & 9 & -4 \\ 6 & -3 & 1 \end{bmatrix} ``` ```math = \begin{bmatrix} 3+(-2) & -1+9 & 7+(-4) \\ 0+6 & 5+(-3) & 2+1 \end{bmatrix} ``` ```math = \begin{bmatrix} 1 & 8 & 3 \\ 6 & 2 & 3 \end{bmatrix} ``` Addition of matrix $$B$$ and $$A$$ ($$B+A$$): Visible text: Addition of matrix and (): Component: MathContainer Children: ```math B+A = \begin{bmatrix} -2 & 9 & -4 \\ 6 & -3 & 1 \end{bmatrix} + \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix} ``` ```math = \begin{bmatrix} -2+3 & 9+(-1) & -4+7 \\ 6+0 & -3+5 & 1+2 \end{bmatrix} ``` ```math = \begin{bmatrix} 1 & 8 & 3 \\ 6 & 2 & 3 \end{bmatrix} ``` (Commutative property proven: $$A+B = B+A$$) Visible text: (Commutative property proven: ) Addition of matrix $$A$$ and $$C$$ ($$A+C$$): Visible text: Addition of matrix and (): Cannot be calculated. Matrix $$A$$ is of order $$2 \times 3$$, while matrix $$C$$ is of order $$2 \times 2$$. Since their orders are different, the addition $$A+C$$ cannot be performed. Visible text: Cannot be calculated. Matrix is of order , while matrix is of order . Since their orders are different, the addition cannot be performed. **Problem** $$2$$ Visible text: **Problem** Given the matrix addition: ```math \begin{bmatrix} 2x & 5 \\ -1 & 3y \end{bmatrix} + \begin{bmatrix} 4 & -2 \\ z & 7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix} ``` Perform matrix addition on the left side: Component: MathContainer Children: ```math \begin{bmatrix} 2x+4 & 5+(-2) \\ -1+z & 3y+7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix} ``` ```math \begin{bmatrix} 2x+4 & 3 \\ -1+z & 3y+7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix} ``` Based on the equality of two matrices, corresponding elements must be equal: For the element in row $$1$$, column $$1$$: $$2x+4 = 10$$ Visible text: For the element in row , column : Component: MathContainer Children: ```math 2x = 10-4 ``` ```math 2x = 6 ``` ```math x = 3 ``` For the element in row $$1$$, column $$2$$: $$3 = 3$$ (already consistent). Visible text: For the element in row , column : (already consistent). For the element in row $$2$$, column $$1$$: $$-1+z = 6$$ Visible text: For the element in row , column : Component: MathContainer Children: ```math z = 6+1 ``` ```math z = 7 ``` For the element in row $$2$$, column $$2$$: $$3y+7 = 1$$ Visible text: For the element in row , column : Component: MathContainer Children: ```math 3y = 1-7 ``` ```math 3y = -6 ``` ```math y = -2 ``` Thus, the values are $$x=3$$, $$y=-2$$, and $$z=7$$. Visible text: Thus, the values are , , and . **Problem** $$3$$ Visible text: **Problem** Given matrix $$P = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix}$$. Visible text: Given matrix . The additive inverse of $$P$$, which is $$-P$$, is: Visible text: The additive inverse of , which is , is: ```math -P = \begin{bmatrix} -1 & -(-2) \\ -3 & -0 \end{bmatrix} = \begin{bmatrix} -1 & 2 \\ -3 & 0 \end{bmatrix} ``` Proof that $$P + (-P) = O$$: Visible text: Proof that : Component: MathContainer Children: ```math P + (-P) = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} -1 & 2 \\ -3 & 0 \end{bmatrix} ``` ```math = \begin{bmatrix} 1+(-1) & -2+2 \\ 3+(-3) & 0+0 \end{bmatrix} ``` ```math = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} ``` The result is the zero matrix $$O$$ of order $$2 \times 2$$. Proven. Visible text: The result is the zero matrix of order . Proven.