Learn what a zero matrix is and how it relates to matrix addition, subtraction, and scalar multiplication.

What you should be familiar with before taking this lesson

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.
If this is new to you, you might want to check out our intro to matrices. You should also make sure you know how to add and subtract matrices and how to multiply a matrix by a scalar.

Definition of zero matrix

A zero matrix is a matrix in which all of the entries are 00. Some examples are given below.
3×33\times 3 zero matrix: O3×3=[000000000]\qquad O_{3\times 3}=\left[\begin{array}{rrr}0 & 0&0 \\ 0 & 0&0 \\ 0 & 0&0 \end{array}\right]
2×42\times 4 zero matrix: O2×4=[00000000]\qquad O_{2\times 4}=\left[\begin{array}{rrrr}0 & 0 &0&0 \\ 0 & 0&0&0 \end{array}\right]
A zero matrix is indicated by OO, and a subscript can be added to indicate the dimensions of the matrix if necessary.
Zero matrices play a similar role in operations with matrices as the number zero plays in operations with real numbers. Let's take a look.

Investigation: What happens when we add a zero matrix?

Recall that to add two matrices, we simply add the corresponding entries.
For example:
[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}
Now try the following matrix addition problems. Notice that each problem involves the sum of a matrix and a zero matrix.
1)
[4513]+[0000]=\left[\begin{array}{rr}{4} &5 \\ 1& 3 \end{array}\right]+\left[\begin{array}{rr}{0} &0 \\ 0& 0 \end{array}\right]=
[4513]+[0000]=[4+05+01+03+0]=[4513]\begin{aligned}\left[\begin{array}{rr}{4} &5 \\ 1& 3 \end{array}\right]+\left[\begin{array}{rr}{0} &0 \\ 0& 0 \end{array}\right]&=\left[\begin{array}{rr}{4+0} &5+0 \\ 1+0& 3+0 \end{array}\right] \\\\ &=\left[\begin{array}{rr}{4} &5 \\ 1& 3 \end{array}\right] \end{aligned}
2)
[000000]+[234817]=\left[\begin{array}{rr}{0} &0 \\ 0& 0\\0&0 \end{array}\right]+\left[\begin{array}{rr}{-2} &3 \\ 4& 8 \\-1&7 \end{array}\right]=
[000000]+[234817]=[0+(2)0+30+40+80+(1)0+7]=[234817]\begin{aligned}\left[\begin{array}{rr}{0} &0 \\ 0& 0\\0&0 \end{array}\right]+\left[\begin{array}{rr}{-2} &3 \\ 4& 8 \\-1&7 \end{array}\right]&=\left[\begin{array}{rr}{0+(-2)} &0+3 \\ 0+4& 0+8 \\0+(-1)&0+7 \end{array}\right]\\ \\ &=\left[\begin{array}{rr}{-2} &3 \\ 4& 8 \\-1&7 \end{array}\right] \end{aligned}

The conclusion

When we add the m×nm\times n zero matrix to any m×nm\times n matrix AA, we get matrix AA back. In other words, A+O=AA+O=A and O+A=AO+A=A.
Here the dimensions of the zero matrix are not explicitly given. It is understood that the dimensions of the zero matrix match the dimensions of matrix AA.

Reflection question

What are the dimensions of the zero matrix in the equation B+O=BB+O=B given that B=[256818]B=\left[\begin{array}{rr}{-2} &5 &6 \\ 8& 1&8 \end{array}\right]?
×\times
B=[256818]B=\left[\begin{array}{rr}{-2} &5 &6 \\ 8& 1&8 \end{array}\right] is a 2×32\times 3 matrix. Because we know B+O=BB+O=B, the addition of BB and the zero matrix is defined. Therefore, OO must have the same dimensions as matrix BB. So OO must be the 2×32\times 3 zero matrix.

Investigation: What happens when we add opposite matrices?

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=[4162]A=\left[\begin{array}{rr}{4} &1 \\ -6& 2 \end{array}\right], then A=[4162]-A=\left[\begin{array}{rr}{-4} &-1 \\ 6& -2 \end{array}\right].
Now try the following matrix addition problems. Notice that each problem involves the sum of a matrix and its opposite.
3)
[4387]+[4387]=\left[\begin{array}{rr}{4} &-3 \\ 8& 7 \end{array}\right]+\left[\begin{array}{rr}{-4} &3 \\ -8& -7 \end{array}\right]=
[4387]+[4387]=[4+(4)3+38+(8)7+(7)]=[0000]\begin{aligned}\left[\begin{array}{rr}{4} &-3 \\ 8& 7 \end{array}\right]+\left[\begin{array}{rr}{-4} &3 \\ -8& -7 \end{array}\right]&=\left[\begin{array}{rr}{4+(-4)} &-3+3 \\ 8+(-8)& 7+(-7)\end{array}\right]\\\\ &=\left[\begin{array}{rr}{0} &0 \\ 0&0 \end{array}\right]\end{aligned}

The conclusion

When we add any m×nm\times n matrix to its opposite, we get the m×nm\times n zero matrix. So if AA is any matrix, then A+(A)=OA+(-A)=O and A+A=O-A+A=O.
It is also true that AA=OA-A=O. This is because subtracting a matrix is like adding its opposite.
Subtracting a matrix is like adding its opposite. In other words, if AA and BB are two matrices, then AB=A+(B)A-B=A+(-B).
The following example justifies this statement.
Suppose A=[2435]A=\left[\begin{array}{rr}{2} &4\\ 3& 5\end{array}\right] and B=[5164]B=\left[\begin{array}{rr}{5} &1\\ 6& 4\end{array}\right].
AB=[2435][5164]=[25413654]=[2+(5)4+(1)3+(6)5+(4)]Subtraction is addition of the opposite=[2435]+[5164]=A+(B)\begin{aligned} A-B &=\left[\begin{array}{rr}{2} &4\\ 3& 5\end{array}\right]-\left[\begin{array}{rr}{5} &1\\ 6& 4\end{array}\right]\\\\ &=\left[\begin{array}{rr}{2-5} &4-1\\ 3-6& 5-4\end{array}\right] \\\\ &=\left[\begin{array}{rr}{2+(-5)} &4+(-1)\\ 3+(-6)& 5+(-4)\end{array}\right]&\small{\gray{\text{Subtraction is addition of the opposite}}}\\\\ &=\left[\begin{array}{rr}{2} &4\\ 3& 5\end{array}\right]+\left[\begin{array}{rr}{-5} &-1\\ -6&-4\end{array}\right]\\\\ &=A+(-B) \end{aligned}

Investigation: What happens when we multiply a matrix by the scalar 00?

When we multiply a matrix by a scalar, each entry in the matrix is multiplied by the given scalar.
When we work with matrices, we refer to real numbers as scalars.
The term scalar multiplication refers to the product of a real number and a matrix.
For example:
2[5231]=[25222321]=[10462]\begin{aligned}\goldD{2}\cdot{\left[\begin{array}{rr}{5} &2 \\ 3& 1 \end{array}\right]}&={\left[\begin{array}{ll}{\goldD2 \cdot5} &\goldD2\cdot 2 \\ \goldD2\cdot3& \goldD2\cdot1 \end{array}\right]}\\\\\\ &={\left[\begin{array}{rr}{10} &4 \\ 6&2 \end{array}\right]}\end{aligned}
Now try the following matrix scalar multiplication problems. Notice that each problem involves multiplying a matrix by the scalar 00.
5)
0[5491]=0\cdot {\left[\begin{array}{rr}{5} &4 \\ 9&1 \end{array}\right]}=
0[5491]=[05040901]=[0000]\begin{aligned}0\cdot {\left[\begin{array}{rr}{5} &4 \\ 9&1 \end{array}\right]}&=\left[\begin{array}{rr}{0\cdot 5} &0\cdot 4 \\ 0\cdot 9&0\cdot 1 \end{array}\right]\\\\ &=\left[\begin{array}{rr}{0} &0 \\ 0&0 \end{array}\right]\end{aligned}
6)
0[2410715342]=0\cdot {\left[\begin{array}{rrr}{-2} &4 &10 \\ 7&-1&5\\-3&4&2 \end{array}\right]}=
0[2410715342]=[0(2)04010070(1)050(3)0402]=[000000000]\begin{aligned}0\cdot {\left[\begin{array}{rrr}{-2} &4 &10 \\ 7&-1&5\\-3&4&2 \end{array}\right]}&=\left[\begin{array}{rrr}{0\cdot (-2)} &0\cdot 4 &0\cdot 10 \\ 0\cdot 7&0\cdot (-1)&0\cdot 5\\0\cdot (-3)&0\cdot 4&0\cdot 2 \end{array}\right]\\\\ &=\left[\begin{array}{rrr}{0} &0 &0 \\ 0&0&0\\0&0&0 \end{array}\right]\end{aligned}

The conclusion

When we multiply any m×nm\times n matrix by the scalar 00, we get the m×nm\times n zero matrix.
Mathematically, this means that 0A=O0A=O.

Summary: Comparing the zero matrix to the real number zero

In the investigations above, we saw that a zero matrix behaves much like the real number zero.
In particular, we can make the following connections:
The number zeroThe zero matrix
Adding zero to any number aa gives back that number aa. (eg. )Adding a zero matrix to any matrix AA gives back the matrix AA. (eg. A+O=O+A=AA+O=O+A=A)
Adding any number to its opposite will give zero. (eg. a+(a)=0a+(-a)=0)Adding any matrix to its opposite will give a zero matrix. (e.g. A+(A)=OA+(-A)=O)
Any number times zero is zero. (e.g a0=0a\cdot 0=0).Scalar multiplication of a matrix by 00 will give a zero matrix. (eg. 0A=O0A=O)
Understanding these connections can help make matrix calculations involving a zero matrix much easier!
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