What is the product of AB in the top question on this page?

Hello! The first thing to do will be to determine the dimensions of our product matrix (I'll call it C). Because matrix A has 3 rows, and matrix B has 2 columns, matrix C will be a 3x2 matrix. 3 rows, 2 columns. Now, the rules for matrix multiplication say that entry i,j of matrix C is the dot product of row i in matrix A and column j in matrix B. We can use this information to find every entry of matrix C. Here are the steps for each entry: Entry 1,1: `(2,4) * (2,8) = 2*2 + 4*8 = 4 + 32 = 36` Entry 2,1: `(6,4) * (2,8) = 6*2 + 4*8 = 12 + 32 = 44` Entry 3,1: `(7,3) * (2,8) = 7*2 + 3*8 = 14 + 24 = 38` Entry 1,2: `(2,4) * (1,5) = 2*1 + 4*5 = 2 + 20 = 22` Entry 2,2: `(6,4) * (1,5) = 6*1 + 4*5 = 6 + 20 = 26` Entry 3,2: `(7,3) * (1,5) = 7*1 + 3*5 = 7 + 15 = 22` Therefore, our final answer is: ``` 36 22 44 26 38 22``` I've left a link to a full tutorial on matrix multiplication at the bottom. Hope this helps! https://www.khanacademy.org/math/precalculus/precalc-matrices/multiplying-matrices-by-matrices/a/multiplying-matrices

can i multiply more than 3 matrices?

You could multiply as many matrices as you like, so long as the order of multiplication and the dimensions of the matrices are such that multiplication is always well-defined. The easiest way to make sure it's well-defined is to multiply a bunch of square matrices of equal dimensions.

how do you find out the dimensions for matrices that are not defined or is it not possible

Not possible. A matrix that is not defined cannot exist; it has no dimensions because there is no solution to the question.

What is the point of ascertaining whether the resultant matrix from a specific ordered multiplication is DEFINED?

Unlike multiplication of numbers or variable, the usual properties of multiplication do not always apply to multiplication of matrices. You need to learn what you can do and what you can't do.

What are the dimensions of a 2 by 3 matrix and a 3 by 2 matrix

Assuming you mean you're multiplying them, the answer would be 2 x 2. You take the number of rows from the first matrix (2) to find the first dimension, and the number of columns from the second matrix (2) to find the second dimension. Another way to think of this: The dimensions of their product is the two outside dimensions. Every time I multiply matrixes on paper, I write the dimensions underneath the matrixes. You can also just do this step mentally. Then, just look at the outer dimensions, and those will be the new dimensions. This helps me remember how to do it. Hope this answered your question.

Main content

Matrix multiplication dimensions

Google Classroom

Learn about the conditions for matrix multiplication to be defined, and about the dimensions of the product of two matrices.

What you should be familiar with before taking this lesson

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.

The dimensions of a matrix give the number of rows and columns of the matrix in that order. Since matrix

A

has

2

rows and

3

columns, it is called a

2 \times 3

matrix.

If this is new to you, we recommend that you check out our intro to matrices.

In matrix multiplication, each entry in the product matrix is the dot product of a row in the first matrix and a column in the second matrix.

\begin{array}{ccccccccc} \vec{b_{1}} & \vec{b_{2}} \\ ↓ & ↓ \\ \begin{matrix} \vec{a_{1}} \to \\ \vec{a_{2}} \to \end{matrix} & [\begin{matrix} 1 \\ 2 \end{matrix} & \begin{matrix} 7 \\ 4 \end{matrix}] & \cdot & [\begin{matrix} 3 \\ 5 \end{matrix} & \begin{matrix} 3 \\ 2 \end{matrix}] & = & [\begin{matrix} \vec{a_{1}} \cdot \vec{b_{1}} \\ \vec{a_{2}} \cdot \vec{b_{1}} \end{matrix} & \begin{matrix} \vec{a_{1}} \cdot \vec{b_{2}} \\ \vec{a_{2}} \cdot \vec{b_{2}} \end{matrix}] \\ A & B & C \end{array}

If this is new to you, we recommend that you check out our matrix multiplication article.

What you will learn in this lesson

We will investigate the relationship between the dimensions of two matrices and the dimensions of their product. Specifically, we will see that the dimensions of the matrices must meet a certain condition for the multiplication to be defined.

When is matrix multiplication defined?

In order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

To see why this is the case, consider the following two matrices:

$A = [\begin{array}{rr} 1 & 3 \\ 2 & 4 \\ 2 & 5 \end{array}]$ ‍ and $B = [\begin{array}{rrrr} 1 & 3 & 2 & 2 \\ 2 & 4 & 5 & 1 \end{array}]$ ‍

To find

A B

, we take the dot product of a row in

A

and a column in

B

. This means that the number of entries in each row of $A$ ‍ must be the same as the number of entries in each column of $B$ ‍.

[Why?]

$A = [\begin{array}{rr} 1 & 3 \\ 2 & 4 \\ 2 & 5 \end{array}]$ ‍ and $B = [\begin{array}{rrrr} 1 & 3 & 2 & 2 \\ 2 & 4 & 5 & 1 \end{array}]$ ‍

Note that if a matrix has two entries in each row, then the matrix has two columns. Similarly, if a matrix has two entries in each column, then it must have two rows.

So, it follows that in order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

Check your understanding

A = [\begin{array}{rr} 2 & 4 \\ 6 & 4 \\ 7 & 3 \end{array}]

and

B = [\begin{array}{rr} 2 & 1 \\ 8 & 5 \end{array}]

Is $A B$ ‍ defined?

[I need help!]

C = [\begin{array}{rrrr} 5 & 3 & 6 & 1 \\ 6 & 8 & 5 & 3 \end{array}]

and

D = [\begin{array}{rrrr} 2 & 1 & 8 \\ 7 & 5 & 5 \end{array}]

Is $C D$ ‍ defined?

[I need help!]

A

is a

4 \times 2

matrix and

B

is a

2 \times 3

matrix.

Is $A B$ ‍ defined?

Is $B A$ ‍ defined?

[I need help!]

Dimension property

The product of an

m \times n

matrix and an

n \times k

matrix is an

m \times k

matrix.

Let's consider the product

A B

, where

A = [\begin{array}{rr} 1 & 3 \\ 2 & 4 \\ 2 & 5 \end{array}]

and

B = [\begin{array}{rrrr} 1 & 3 & 2 & 2 \\ 2 & 4 & 5 & 1 \end{array}]

From above, we know that

A B

is defined since the number of columns in

A_{3 \times 2}

(2)

matches the number of rows in

B_{2 \times 4}

(2)

To find

A B

, we must be sure to find the dot product between each row in

A

and each column in

B

. So, the resulting matrix will have the same number of rows as matrix

A_{3 \times 2}

(3)

and the same number of columns as matrix

B_{2 \times 4}

(4)

. It will be a

3 \times 4

matrix.

\begin{array}{ccccccccc} \vec{b_{1}} & \vec{b_{2}} & \vec{b_{3}} & \vec{b_{4}} \\ ↓ & ↓ & ↓ & ↓ \\ \begin{matrix} \vec{a_{1}} \to \\ \vec{a_{2}} \to \\ \vec{a_{3}} \to \end{matrix} & [\begin{matrix} 1 \\ 2 \\ 2 \end{matrix} & \begin{matrix} 3 \\ 4 \\ 5 \end{matrix}] & \cdot & [\begin{matrix} 1 \\ 2 \end{matrix} & \begin{matrix} 3 \\ 4 \end{matrix} & \begin{matrix} 2 \\ 5 \end{matrix} & \begin{matrix} 2 \\ 1 \end{matrix}] & = & [\begin{matrix} \vec{a_{1}} \cdot \vec{b_{1}} \\ \vec{a_{2}} \cdot \vec{b_{1}} \\ \vec{a_{3}} \cdot \vec{b_{1}} \end{matrix} & \begin{matrix} \vec{a_{1}} \cdot \vec{b_{2}} \\ \vec{a_{2}} \cdot \vec{b_{2}} \\ \vec{a_{3}} \cdot \vec{b_{2}} \end{matrix} & \begin{matrix} \vec{a_{1}} \cdot \vec{b_{3}} \\ \vec{a_{2}} \cdot \vec{b_{3}} \\ \vec{a_{3}} \cdot \vec{b_{3}} \end{matrix} & \begin{matrix} \vec{a_{1}} \cdot \vec{b_{4}} \\ \vec{a_{2}} \cdot \vec{b_{4}} \\ \vec{a_{3}} \cdot \vec{b_{4}} \end{matrix}] \\ A & B & A B \end{array}

Check your understanding

A = [\begin{array}{rr} 2 & 4 \\ 6 & 4 \\ 7 & 3 \end{array}]

and

B = [\begin{array}{rr} 2 & 1 \\ 8 & 5 \end{array}]

What are the dimensions of $A B$ ‍?

\times

[I need help!]

C = [\begin{array}{rr} 4 & 3 & 1 \\ 6 & 7 & 2 \end{array}]

and

D = [\begin{array}{r} 3 \\ 1 \\ 4 \end{array}]

What are the dimensions of $C D$ ‍?

\times

[I need help!]

A

is a

2 \times 3

matrix and

B

is a

3 \times 4

matrix.

What are the dimensions of $A B$ ‍?

\times

[I need help!]

X

is a

2 \times 1

matrix and

Y

is a

1 \times 2

matrix.

What are the dimensions of matrix $X Y$ ‍?

\times

[I need help!]

Want to join the conversation?

Sort by:

ranibahuthai
Posted 6 years ago. Direct link to ranibahuthai's post “What is the product of AB...”
What is the product of AB in the top question on this page?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Tucker Quering
  Posted 6 years ago. Direct link to Tucker Quering's post “Hello! The first thing t...”
  Hello!
  
  The first thing to do will be to determine the dimensions of our product matrix (I'll call it C). Because matrix A has 3 rows, and matrix B has 2 columns, matrix C will be a 3x2 matrix. 3 rows, 2 columns.
  
  Now, the rules for matrix multiplication say that entry i,j of matrix C is the dot product of row i in matrix A and column j in matrix B. We can use this information to find every entry of matrix C. Here are the steps for each entry:
  
  Entry 1,1: (2,4) * (2,8) = 2*2 + 4*8 = 4 + 32 = 36
  Entry 2,1: (6,4) * (2,8) = 6*2 + 4*8 = 12 + 32 = 44
  Entry 3,1: (7,3) * (2,8) = 7*2 + 3*8 = 14 + 24 = 38
  Entry 1,2: (2,4) * (1,5) = 2*1 + 4*5 = 2 + 20 = 22
  Entry 2,2: (6,4) * (1,5) = 6*1 + 4*5 = 6 + 20 = 26
  Entry 3,2: (7,3) * (1,5) = 7*1 + 3*5 = 7 + 15 = 22
  
  Therefore, our final answer is:
  
  36 22 44 26 38 22
  
  I've left a link to a full tutorial on matrix multiplication at the bottom. Hope this helps!
  
  https://www.khanacademy.org/math/precalculus/precalc-matrices/multiplying-matrices-by-matrices/a/multiplying-matrices
  Comment on Tucker Quering's post “Hello! The first thing t...”
  (22 votes)
Racher
Posted 6 years ago. Direct link to Racher's post “can i multiply more than ...”
can i multiply more than 3 matrices?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- kubleeka
  Posted 6 years ago. Direct link to kubleeka's post “You could multiply as man...”
  You could multiply as many matrices as you like, so long as the order of multiplication and the dimensions of the matrices are such that multiplication is always well-defined. The easiest way to make sure it's well-defined is to multiply a bunch of square matrices of equal dimensions.
  Button navigates to signup page
  (12 votes)
gtc1021
Posted 7 years ago. Direct link to gtc1021's post “how do you find out the d...”
how do you find out the dimensions for matrices that are not defined or is it not possible
Button navigates to signup pageComment on gtc1021's post “how do you find out the d...”
(1 vote)
Answer
- steven.m.klassen
  Posted 7 years ago. Direct link to steven.m.klassen's post “Not possible. A matrix th...”
  Not possible.
  A matrix that is not defined cannot exist; it has no dimensions because there is no solution to the question.
  Comment on steven.m.klassen's post “Not possible. A matrix th...”
  (12 votes)
Olivia Nesbitt
Posted 6 years ago. Direct link to Olivia Nesbitt's post “Why (like in questions 3 ...”
Why (like in questions 3 & 4) is AB defined but BA is not defined? Do matrices not conform to commutative properties of multiplication?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “Correct... the commutativ...”
  Correct... the commutative property of multiplication does not work with matrices. Go back and pay close attention to the section above titled: When is Matrix Multiplication Defined?". The last line states: "So, it follows that in order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix". The 2 questions you are asking about do not meet this condition which is why the multiplication is not defined.
  
  Hope this helps.
  Button navigates to signup page
  (8 votes)
Viacheslav VOROBIEFF
Posted 5 years ago. Direct link to Viacheslav VOROBIEFF's post “What is the point of asce...”
What is the point of ascertaining whether the resultant matrix from a specific ordered multiplication is DEFINED?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Kim Seidel
  Posted 5 years ago. Direct link to Kim Seidel's post “Unlike multiplication of ...”
  Unlike multiplication of numbers or variable, the usual properties of multiplication do not always apply to multiplication of matrices. You need to learn what you can do and what you can't do.
  Button navigates to signup page
  (5 votes)
Olly
Posted 5 years ago. Direct link to Olly's post “how can I learn the non-z...”
how can I learn the non-zero value in matrix
Button navigates to signup pageComment on Olly's post “how can I learn the non-z...”
(3 votes)
Answer
- SnappyRiffs
  Posted 6 months ago. Direct link to SnappyRiffs's post “A non-zero value in a mat...”
  A non-zero value in a matrix is where the matrix does not have a zero in it.
  Button navigates to signup page
  (1 vote)
mary nakombe
Posted 6 years ago. Direct link to mary nakombe's post “How can I find the invers...”
How can I find the inverse of a 3*3?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- SnappyRiffs
  Posted 6 months ago. Direct link to SnappyRiffs's post “The following video gives...”
  The following video gives the idea of how to find the inverse of a matrix: https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-intro-to-matrix-inverses/v/inverse-matrix-part-1
  Button navigates to signup page
  (1 vote)
19jbenoit
Posted 6 years ago. Direct link to 19jbenoit's post “What are the dimensions o...”
What are the dimensions of a 2 by 3 matrix and a 3 by 2 matrix
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- Annika
  Posted 6 years ago. Direct link to Annika's post “Assuming you mean you're ...”
  Assuming you mean you're multiplying them, the answer would be 2 x 2. You take the number of rows from the first matrix (2) to find the first dimension, and the number of columns from the second matrix (2) to find the second dimension.
  Another way to think of this:
  The dimensions of their product is the two outside dimensions. Every time I multiply matrixes on paper, I write the dimensions underneath the matrixes. You can also just do this step mentally. Then, just look at the outer dimensions, and those will be the new dimensions. This helps me remember how to do it.
  
  Hope this answered your question.
  Button navigates to signup page
  (4 votes)
amana.cla
Posted 3 months ago. Direct link to amana.cla's post “how can i multiply a matr...”
how can i multiply a matrix of different sizes
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Krithin Anishwa Joka
  Posted 3 months ago. Direct link to Krithin Anishwa Joka's post “"multiplying matrices" se...”
  "multiplying matrices" search this in khan academy they uploaded a video about it. You will understand it there
  Button navigates to signup page
  (1 vote)

Algebra (all content)

Course: Algebra (all content) > Unit 20

Matrix multiplication dimensions

What you should be familiar with before taking this lesson

What you will learn in this lesson

When is matrix multiplication defined?

Check your understanding

Dimension property

Check your understanding

Want to join the conversation?