Learn about the properties of matrix addition (like the commutative property) and how they relate to real number addition.
In the table below, AA, BB, and CC are matrices of equal dimensions.
PropertyExample
Commutative property of additionA+B=B+A{A}+{B}={B}+{A}
Associative property of additionA+(B+C)=(A+B)+C{A}+({B}+{C})=({A}+{B})+{C}
Additive identity propertyFor any matrix AA, there is a unique matrix OO such that A+O=AA+O=A.
Additive inverse propertyFor each AA, there is a unique matrix A-A such that A+(A)=OA+(-A)=O.
Closure property of additionA+BA+B is a matrix of the same dimensions as AA and BB.
This article explores these matrix addition properties.

Matrices and matrix addition

A matrix is a rectangular arrangement of numbers into rows and columns. The dimensions of a matrix give the number of rows and columns of the matrix in that order. Since matrix AA has 22 rows and 33 columns, it is called a 2×32\times 3 matrix.
To add two matrices of the same dimensions, simply add the entries in the corresponding positions.
[3724]+[5281]=[3+57+22+84+1]=[89105]\begin{aligned}{\left[\begin{array}{rr}{\blueD3} &\blueD7 \\ \blueD2& \blueD4 \end{array}\right]}+\left[\begin{array}{rr}{\greenD5} &\greenD2 \\ \greenD8& \greenD1 \end{array}\right]&={\left[\begin{array}{rr}{\blueD3+\greenD5} &\blueD7+\greenD2 \\\blueD2+\greenD8& \blueD4+\greenD1 \end{array}\right]} \\\\ &=\left[\begin{array}{rr}{8} &9 \\ 10& 5 \end{array}\right]\\ \end{aligned}
If any of this is new to you, you should check out the following articles before you proceed:

Dimensions considerations

Notice that the sum of two 2×22\times 2 matrices is another 2×22\times 2 matrix. In general, the sum of two m×n m\times n matrices is another m×nm\times n matrix. This describes the closure property of matrix addition.
If the dimensions of two matrices are not the same, the addition is not defined. This is because if AA is a 2×32\times 3 matrix and BB is a 2×22\times 2 matrix, then some entries in matrix AA will not have corresponding entries in matrix BB!
[278243]+[5281]=undefined\begin{aligned}{\left[\begin{array}{rr}\blueD2&\blueD7 &\goldD8 \\ \blueD2& \blueD4&\goldD3 \end{array}\right]}+\left[\begin{array}{rr}{\greenD5} &\greenD2 \\ \greenD8& \greenD1 \end{array}\right]&=\text{undefined} \end{aligned}

Matrix addition & real number addition

Because matrix addition relies heavily on the addition of real numbers, many of the addition properties that we know to be true with real numbers are also true with matrices.
Let's take a look at each property individually.

Commutative property of addition: A+B=B+AA+B=B+A

This property states that you can add two matrices in any order and get the same result.
This parallels the commutative property of addition for real numbers. For example, 3+5=5+33+5=5+3.
The following example illustrates this matrix property.
Notice how the commutative property of addition for matrices holds thanks to the commutative property of addition for real numbers!

Associative property of addition: (A+B)+C=A+(B+C)(A+B)+C=A+(B+C)

This property states that you can change the grouping in matrix addition and get the same result. For example, you can add matrix AA to BB first, and then add matrix CC, or, you can add matrix BB to CC, and then add this result to AA.
This property parallels the associative property of addition for real numbers. For example, (2+3)+5=2+(3+5)(2+3)+5=2+(3+5).
Let's justify this matrix property by looking at an example.
In each column we simplified one side of the identity into a single matrix. The two resulting matrices are equivalent thanks to the real number associative property of addition. For example, (5+3)+1=5+(3+1)(\tealD5+\purpleC3)+\goldD1=\tealD 5+(\purpleC 3+\goldD1).
Because of this property, we can write down an expression like A+B+CA+B+C and have this be completely defined. We do not need parentheses indicating which addition to perform first, as it doesn't matter!

Additive identity property: A+O=AA+O=A

A zero matrix, denoted OO, is a matrix in which all of the entries are 00.
Notice that when a zero matrix is added to any matrix AA, the result is always AA.
  • [3179]+[0000]=[3179]\left[\begin{array}{rr}{3} &-1 \\ 7& 9 \end{array}\right]+\left[\begin{array}{rr}{0} &0 \\ 0& 0 \end{array}\right]=\left[\begin{array}{rr}{3} &-1 \\ 7& 9 \end{array}\right]
  • [000000]+[283157]=[283157]\left[\begin{array}{rr}{0} &0 &{0} \\ 0& 0&{0} \end{array}\right]+\left[\begin{array}{rr}{2} &8 &3 \\ -1& 5&7 \end{array}\right]=\left[\begin{array}{rr}{2} &8 &3 \\ -1& 5&7 \end{array}\right]
These examples illustrate what is meant by the additive identity property; that the sum of any matrix AA and the the appropriate zero matrix is the matrix AA.
A zero matrix can be compared to the number zero in the real number system. For all real numbers aa, we know that a+0=aa+0=a. The number 00 is the additive identity in the real number system just like OO is the additive identity for matrices.

Additive inverse property: A+(A)=OA+(-A)=O

The opposite of a matrix AA is the matrix A-A, where each element in this matrix is the opposite of the corresponding element in matrix AA.
For example, if A=[2831]A=\left[\begin{array}{rr}{-2} &8 \\ -3& 1 \end{array}\right], then A=[2831]-A=\left[\begin{array}{rr}{2} &-8 \\ 3& -1 \end{array}\right].
If we add AA to A-A we get a zero matrix, which illustrates the additive inverse property.
A+(A)=[2831]+[2831]=[2+28+(8)3+31+(1)]=[0000]\begin{aligned}A+(-A)&=\left[\begin{array}{rr}{-2} &8 \\ -3& 1 \end{array}\right]+\left[\begin{array}{rr}{2} &-8 \\ 3& -1 \end{array}\right]\\\\&=\left[\begin{array}{rr}{-2+2} &8+(-8) \\ -3+3& 1+(-1) \end{array}\right]\\\\ &=\left[\begin{array}{rr}{0} &0 \\ 0& 0 \end{array}\right] \end{aligned}
The sum of a real number and its opposite is always 00, and so the sum of any matrix and its opposite gives a zero matrix. Because of this, we refer to opposite matrices as additive inverses.

Check your understanding

For the problems below, let AA, BB, and CC be 2×22\times 2 matrices.
1) Which of the following matrix expressions are equivalent to (A+B)+C?(A+B)+C?
Choose all answers that apply:
Choose all answers that apply:

Because matrix addition is associative, we know that (A+B)+C=A+(B+C)(A+B)+C=A+(B+C).
Since the grouping of an addition problem doesn't matter, the above expressions are also equal to A+B+CA+B+C.
Using the commutative and associative properties of matrix addition, we can also show that (A+B)+C=(C+A)+B(A+B)+C=(C+A)+B.
(A+B)+C=A+(B+C)Associative property of matrix addition=A+(C+B)Commutative property of matrix addition=(A+C)+BAssociative property of matrix addition=(C+A)+BCommutative property of matrix addition\begin{aligned}(A+B)+C&=A+(B+C)&\small{\gray{\text{Associative property of matrix addition}}}\\ \\ &=A+(C+B)&\small{\gray{\text{Commutative property of matrix addition}}}\\ \\ &=(A+C)+B&\small{\gray{\text{Associative property of matrix addition}}}\\ \\ &=(C+A)+B&\small{\gray{\text{Commutative property of matrix addition}}} \end{aligned}
(A+C)+(B+C)(A+C)+(B+C) is not equivalent to the others, since there is an extra CC added to the expression.
The following expressions are equivalent to (A+B)+C(A+B)+C:
  • A+(B+C)A+(B+C)
  • (C+A)+B(C+A)+B
  • A+B+CA+B+C
2) Which of the following matrix expressions are equivalent to (A+(A))+B(A+(-A))+B?
Remember AA and BB are 2×22\times 2 matrices.
Choose all answers that apply:
Choose all answers that apply:

Because of the additive inverse and additive identity properties, we know:
(A+(A))+B=O+B=B\begin{aligned}(A+(-A))+B&=O+B\\ \\ &=B \end{aligned}
This is also equivalent to [0000]+B\left[\begin{array}{rr}{0} &0 \\ 0& 0 \end{array}\right]+B, since we know that BB is a 2×22\times 2 matrix.
Finally, we know that (A+(A))+B=A+(A+B)(A+(-A))+B=A+(-A+B) by the associative property of matrix addition.
The following expressions are equivalent to (A+(A))+B(A+(-A))+B:
  • BB
  • A+(A+B)A+(-A+B)
  • [0000]+B\left[\begin{array}{rr}{0} &0 \\ 0& 0 \end{array}\right]+B
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