Compatible Matrices
When adding and subtracting matrices, each matrix must have the same dimensions. But when multiplying matrices, they must instead be compatible, which means the number of columns in the first matrix must be equal to the number of rows in the second matrix. Let''s take a closer look at compatible matrices.
Compatible matrices: Definition and multiplication
Compatible matrices are matrices that can be multiplied; the number of columns in the first matrix equals the number of rows in the second matrix.
Let''s look at the following examples:
Example #1
Matrix $A$ is:
$\left[\begin{array}{ccc}2& 1& 3\end{array}\right]$Matrix $B$ is:
$\left[\begin{array}{cc}2& 1\\ 3& 4\\ 1& 2\end{array}\right]$In this example, the number of columns in the first matrix (3) equals the number of rows in the second matrix (3), so we know these two matrices are compatible and can be multiplied.
Example #2
Matrix $A$ is:
$\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$Matrix $B$ is:
$\left[\begin{array}{cc}2& 1\\ 3& 4\\ 1& 2\end{array}\right]$In this example, the number of columns in the first matrix (2) does not equal the number of rows in the second matrix (3). Therefore, these matrices are not compatible.
What if we''d like to multiply the compatible matrices in Example #1?
First, we need to find the dimensions of the product. The product of an $\left(a\times b\right)$ matrix and $a\left(b\times c\right)$ matrix has dimensions $\left(a\times c\right)$ . So, in this case, the product of the $1\times 3$ matrix (Matrix $A$ ) and the $3\times 2$ matrix (Matrix $B$ ) has dimensions $1\times 2$ .
Now, let''s multiply the matrices.
$A\times B=\left[\begin{array}{ccc}2& 1& 3\end{array}\right]\times \left[\begin{array}{cc}2& 1\\ 3& 4\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}2\times 2+1\times 3+3\times 1& 2\times 1+1\times 4+3\times 2\end{array}\right]=\left[\begin{array}{cc}10& 12\end{array}\right]$So, the product of Matrix $A$ and Matrix $B$ is a $1\times 2$ matrix: $\left[\begin{array}{cc}10& 12\end{array}\right]$ .
Practice questions on compatible matrices

True or false: For compatible matrices, the number of columns in
the first matrix equals the number of rows in the second matrix.
Answer: True 
What are the dimensions of the product of a
$2\times 4$
matrix and a
$4\times 7$
matrix?
Answer: $2\times 7$ 
Is a
$4\times 6$
matrix compatible with a
$5\times 4$
matrix?
Answer: No, the number of columns in the first matrix does not equal the number of rows in the second matrix. 
Is a
$3\times 5$
matrix compatible with a
$5\times 7$
matrix?
Answer: Yes, the number of columns in the first matrix equals the number of rows in the second matrix. 
Multiply the following compatible matrices:
$\left[\begin{array}{ccc}1& 3& 2\end{array}\right]\times \left[\begin{array}{c}5\\ 2\\ 1\end{array}\right]$
The matrices have dimensions of $1\times 3$ and $3\times 1$ , so the dimensions of the product are $1\times 1$ .
Multiply the corresponding numbers for the row and column. Add the products. $1\left(5\right)+3\left(2\right)+2\left(1\right)=5+6+2=13$ Answer: $\left[\begin{array}{c}13\end{array}\right]$
Topics related to the Compatible Matrices
Multiplying Vector by a Matrix
Flashcards covering the Compatible Matrices
Practice tests covering the Compatible Matrices
Gain greater insight into compatible matrices
Compatible matrices can be challenging to grasp, and understanding how to multiply matrices can be even more difficult when doing so for the first time. Working alongside a tutor is a great way to gain clarity. A qualified personal educator can answer any questions whether about finding the dimensions of the product or determining which numbers in each row and column should be multiplied. Tutors can also provide valuable assistance with assignments and test preparations. Get more information about the benefits of tutoring by contacting the Educational Directors at Varsity Tutors today.
 AWS Certified Developer Test Prep
 US History Tutors
 TACHS Test Prep
 Finite Mathematics Tutors
 ACT Math Courses & Classes
 HSPT Test Prep
 PS Exam  Professional Licensed Surveyor Principles of Surveying Exam Courses & Classes
 Elementary School Writing Tutors
 Functions Tutors
 Information Security Tutors
 Certified Information Systems Auditor Test Prep
 Differential Equations Tutors
 ACT Writing Test Prep
 12th Grade Reading Tutors
 Graduate Program Application Essay Tutors
 International Studies Tutors
 SAT Subject Test in United States History Tutors
 Special Needs Tutors
 OLSAT Test Prep
 Marine Science Tutors