If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Multiplying matrices by scalars

Learn how to find the result of a matrix multiplied by a real number.

What you should be familiar with before taking this lesson

3 columns2 rows[255267]\begin{array}{c} \goldE{\text{3 columns}} \\\\ \begin{array}{c} \blueE{\text{2 rows}}&\goldE{\LARGE\downarrow}&\goldE{\LARGE\downarrow}&\goldE{\LARGE\downarrow} \\\\ \begin{array}{c} \blueE{\LARGE\rightarrow} \\\\ \blueE{\LARGE\rightarrow}\end{array} &\left[\begin{array}{c} -2 \\\\ 5\end{array}\right. &\begin{array}{c}5 \\\\ 2\end{array} &\left.\begin{array}{c}6 \\\\ 7\end{array}\right] \end{array} \end{array}
A matrix is a rectangular arrangement of numbers into rows and columns. Each number in a matrix is referred to as a matrix element or entry.
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.

What you will learn in this lesson

We can multiply matrices by real numbers. This article explores how this works.

Scalars and scalar multiplication

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.
For example, given that A=[10643]\bold A=\left[\begin{array}{c} 10 &6 \\\\ 4& 3 \end{array}\right], let's find 2, A.
To find 2, A, simply multiply each matrix entry by 2:
2A=2[10643]=[210262423]=[201286]\begin{aligned} \greenD 2\bold A&=\greenD{2}\cdot{\left[\begin{array}{c} 10 &6 \\\\ 4& 3 \end{array}\right]} \\\\ &={\left[\begin{array}{c} \greenD2 \cdot10 &\greenD2\cdot 6 \\\\ \greenD2\cdot 4& \greenD2\cdot3 \end{array}\right]} \\\\ &=\left[\begin{array}{c} 20 &12 \\\\ 8& 6 \end{array}\right] \end{aligned}

Check your understanding

Problem 1
Given B=[4271]\bold B=\left[\begin{array}{c} -4 &-2 \\\\ 7& 1 \end{array}\right], find minus, 3, B.
minus, 3, B, equals

Problem 2
Given C=[42273]\bold C=\left[\begin{array}{c} -42 \\\\ 27 \\\\ -3 \end{array}\right], find start fraction, 1, divided by, 3, end fraction, C.
start fraction, 1, divided by, 3, end fraction, C, equals

Want to join the conversation?

  • leafers seedling style avatar for user Rishikesh
    What about scalar division, ie division of a matrix by a real number? Will the same rules apply as in scalar multiplication?
    (17 votes)
    Default Khan Academy avatar avatar for user
    • aqualine sapling style avatar for user Jasper Williams
      Short answer - yes, Absolutely!

      Longer answer - You can view scalar division as multiplying by the reciprocal [i.e dividing a number/matrix by a set number is the same as multiplying by 1/number]
      For example: 15/3 = 15*1/3.
      Hence if you want to divide a matrix by a scalar simply multiply the matrix by the reciprocal of your divider (or just divide, its the same thing)
      (36 votes)
  • duskpin ultimate style avatar for user Violamaster
    what would a matrice multiplied by a zero be?
    (6 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Deepam Sarmah
    In the definition of scalar multiplication the scalar must be a real number. But what happens when that scalar is an imaginary number? Will the multiplication of the imaginary number with the matrix be defined as undefined?
    (10 votes)
    Default Khan Academy avatar avatar for user
  • duskpin seedling style avatar for user allison.helton
    I was solving a Khan Academy problem where I had to solve for X. There were two matrices of equal dimensions (3x2) and X is in one of them. I had to add the matrices together and multiply by 2. How come it displays the answer as a 3x3 dimensional matrix when the answer is actually a 3x2 matrix? I looked at the help section and it stated that the answer is actually a 3x2 answer.
    (7 votes)
    Default Khan Academy avatar avatar for user
  • piceratops tree style avatar for user MoulikLaddha
    what are scalar numbers?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • spunky sam blue style avatar for user ADITYA
    can anyone explain clearly with example on how to do fractional scalar division?
    (6 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Iza
      With a fractional scalar division you would kind of divide the numbers for example - let the scalar be 1/3 and the number in the matrix be 12 you would just put a 1 under the 12 to make it a fraction, then just multiply, giving you 4.
      (2 votes)
  • leaf red style avatar for user layaz7717
    So scalar division exists too, and we can use it, but we more often just multiply by the reciprocal, correct?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Mike
    I don't see how substitution is helpful...it adds an extra step. You still have to do the work of merging the A and B matrix together in one way or another. 1/3X + A = B....why not just skip renaming them A and B and just subtract the left side matrix from the right side, then scale up by 3?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Alishah Ahmad
    In a matrix how would you multiply pi and a scaler
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user zaitoumessaoud
    I cant solve the matrics equations
    (2 votes)
    Default Khan Academy avatar avatar for user