Learn about the properties of matrix scalar multiplication (like the distributive property) and how they relate to real number multiplication.
In the table below, AA and BB are matrices of equal dimensions, cc and dd are scalars, and OO is a zero matrix.
PropertyExample
Associative property of multiplication(cd)A=c(dA)(cd)A=c(dA)
Distributive properties c(A+B)=cA+cBc(A+B)=cA+cB
(c+d)A=cA+dA(c+d)A=cA+dA
Multiplicative identity property 1A=A1 A=A
Multiplicative properties of zero0A=O0\cdot A=O
cO=Oc\cdot O=O
Closure property of multiplicationcAcA is a matrix of the same dimensions as AA.
This article explores these properties.

Matrices and scalar multiplication

A matrix is a rectangular arrangement of numbers into rows and columns.
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. In scalar multiplication, each entry in the matrix is multiplied by the given scalar.
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}
If any of this is new to you, you should check out the following articles before you proceed:

Dimensions considerations

Notice that a scalar times a 2×22\times 2 matrix is another 2×22\times 2 matrix. In general, a scalar multiple of a matrix will be another matrix of the same dimension. This is what is meant by the closure property of scalar multiplication!

Matrix scalar multiplication & real number multiplication

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

Associative property of multiplication: (cd)A=c(dA)(cd)A=c(dA)

This property states that if a matrix is multiplied by two scalars, you can multiply the scalars together first, and then multiply by the matrix. Or you can multiply the matrix by one scalar, and then the resulting matrix by the other.
The following example illustrates this property for c=2\blueD c=\blueD2, d=3\goldD d=\goldD 3, and A=[5481]\greenD A={\left[\begin{array}{rr}{\greenD5}&\greenD4 \\ \greenD8& \greenD1\end{array}\right]}.
In each column we simplified one side of the identity into a single matrix. Notice that these two matrices are equal because of the associative property of multiplication for real numbers. For example, (23)5=2(35)(\blueD2\cdot \goldD3)\cdot \greenD 5=\blueD 2\cdot(\goldD 3\cdot \greenD5).
This shows that the original expressions must be equivalent as well!

Distributive properties:

c(A+B)=cA+cBc(A+B)=cA+cB

This property states that a scalar can be distributed over matrix addition.
Here's an example where c=2\blueD c=\blueD2, A=[5231]\greenD A=\left[\begin{array}{rr}{\greenD5} &\greenD2\\ \greenD3& \greenD1\end{array}\right], and B=[3426]\goldD B=\left[\begin{array}{rr}{\goldD3} &\goldD4\\ \goldD2& \goldD6\end{array}\right]:
If we compare the last matrix in each column, we see that these are equivalent because of the distributive property for real numbers. For example, 2(5+3)=25+23\blueD2(\greenD5+\goldD3)=\blueD2 \cdot \greenD5+\blueD2 \cdot \goldD3.
Thus the original two expressions must be equivalent as well!

(c+d)A=cA+dA(c+d)A=cA+dA

This property states that a matrix can be distributed over scalar addition.
Here's an example where c=2\blueD c=\blueD2, d=3\greenD d=\greenD3, and A=[6974]\goldD A=\left[\begin{array}{rr}{\goldD6} &\goldD9 \\ \goldD7&\goldD4\end{array}\right]:
Once again, we see that the last matrix in each column are equivalent because of the distributive property for real numbers, making the original expressions equivalent as desired!

Multiplicative identity property: 1A=A1A=A

This property says that when you multiply any matrix AA by the scalar 11, the result is simply the original matrix AA.
So, for example, if A=[2517]A= \left[\begin{array}{rr}{2} &5 \\ 1&7 \end{array}\right], then we have:
1[2517]=[12151117]=[2517]\begin{aligned}\greenD1 \left[\begin{array}{rr}{2} &5 \\ 1&7 \end{array}\right]&=\left[\begin{array}{rr}{\greenD1\cdot {2}} &\greenD1\cdot 5 \\\greenD1\cdot 1&\greenD1\cdot 7 \end{array}\right]\\ \\ &=\left[\begin{array}{rr}{2} &5 \\ 1&7 \end{array}\right] \end{aligned}
Notice that because 1a=a1\cdot a=a for any real number aa, the scalar 11 will always be the multiplicative identity in scalar multiplication!

Multiplicative properties of zero:

0A=O0\cdot A=O

This property states that in scalar multiplication, 00 times any m×nm\times n matrix AA is the m×nm\times n zero matrix.
This is true because of the multiplicative properties of zero in the real number system. If aa is a real number, we know 0a=00\cdot a=0. The following example illustrates this.
0[3867]=[03080607]=[0000]\begin{aligned}\greenD0 \left[\begin{array}{rr}{3} &8 \\ 6&7 \end{array}\right]&=\left[\begin{array}{rr}{\greenD0\cdot 3} &\greenD0\cdot 8 \\ \greenD0\cdot 6&\greenD0\cdot 7 \end{array}\right]\\ \\ &=\left[\begin{array}{rr}{0} &0 \\ 0&0 \end{array}\right] \end{aligned}

cO=Oc\cdot O=O

This property states that any scalar times a zero matrix is the same zero matrix.
Again, this property is true because of the multiplicative properties of zero in the real number system. Here's an example where c=3c=3 and OO is the 2×22\times 2 zero matrix.
3[0000]=[30303030]=[0000]\begin{aligned}\greenD 3 \left[\begin{array}{rr}{0} &0 \\ 0&0 \end{array}\right]&=\left[\begin{array}{rr}{\greenD 3 \cdot 0} &\greenD 3\cdot 0 \\ \greenD 3\cdot 0&\greenD 3\cdot 0 \end{array}\right]\\ \\ &=\left[\begin{array}{rr}{0} &0 \\ 0&0 \end{array}\right] \end{aligned}

Check your understanding

Now that you are familiar with all of the scalar multiplication properties, let's see if you can use them to determine equivalent matrix expressions.
For the problems below, let AA and BB be 2×22\times 2 matrices and let cc and dd be scalars.
1) Which of the following are equivalent to c(1A+B)c(1A+B)?
Choose all answers that apply:
Choose all answers that apply:

We are given the expression c(1A+B)c(1A+B).
We can use the matrix properties to find equivalent expressions.
c(1A+B)=c(A+B)Multiplicative identity property=cA+cBDistributive property=cB+cACommutative property of addition\begin{aligned} c(1A+B)&=c(A+B)&\small{\gray{\text{Multiplicative identity property}}}\\ \\ &=\maroonC{cA+cB}&\small{\gray{\text{Distributive property}}}\\ \\ &=\maroonC{cB+cA}&\small{\gray{\text{Commutative property of addition}}} \end{aligned}
These are the equivalent expressions:
  • cA+cBcA+cB
  • cB+cAcB+cA
2) Which of the following are equivalent to (cd)A+0A(cd)A+0A?
Choose all answers that apply:
Choose all answers that apply:

We are given the expression (cd)A+0A(cd)A+0A.
We can use the matrix properties to find equivalent expressions.
(cd)A+0A=(cd)A+OMultiplicative property of zero=(cd)AAdditive identity property=c(dA)Associative property of multiplication\begin{aligned} (cd)A+0A&=(cd)A+O&\small{\gray{\text{Multiplicative property of zero}}}\\ \\ &={(cd)A}&\small{\gray{\text{Additive identity property}}}\\ \\ &=\maroonC{c(dA)}&\small{\gray{\text{Associative property of multiplication}}} \end{aligned}
We also know by the distributive property that (cd)A+0A=(cd+0)A(cd)A+0A=\maroonC{(cd+0)A}.
These are the equivalent expressions:
  • c(dA)c(dA)
  • (cd+0)A(cd+0)A
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